essay on how logarithms used in real life

10 Common Applications Of Logarithms In Real-life

Mathematics is challenging yet fun. To understand mathematical concepts, one should be curious, open-minded, critical thinkers, and have knowledge of inductive and deductive reasoning in mathematics. Logarithms is a challenging mathematical concept, yet children find it easy to solve and apply due to its practicality.

Logarithms have been used for years and have reinforced various scientific discoveries over time. From early astronomical experiments to developing the first data storage device, concepts of exponents and logarithms are widely applied. Considering the same significance of the concept, in this article, we will cover some major real-life applications of logarithms.

Let’s get started!

Understanding the concept of Logarithms

Logarithms is one of the oldest mathematical concepts that made various early scientific and mathematical discoveries possible. The history of logarithms dates back to the 17th century when a Scottish mathematician John Napier invented the concept of logarithms. He coined the term from the Greek words ‘logos’ and ‘arithmos’, which means ‘ratio’ and ‘number’.

Logarithms is a method to represent a significantly large mathematical value, and it helps to identify how many times a number has to be multiplied to get a desired other number. Logarithms are represented as a base number and power to that base number, using the equation logₕ a=y.

Let’s understand logarithms using an example.

Suppose you are a biologist interested in a rare species of bacteria. During your study, you noticed this rare form of bacteria’s population, which is 100, currently, increases at least ten times in an hour. You are interested in finding out how long it will take for the population to reach 10,000. Using logarithms, you can form the equation and find the required time.

Log₁₀ 100 = y

Here, ‘y’ will be the required time, ‘h’ is the rate of increment in population, and ‘a’ is the current population. The value of ‘y’ can be easily found with the help of the ‘log table’.

Application of logarithms in real-life

Logarithms form a base of various scientific and mathematical procedures. Logarithms have wide practicality in solving calculus, statistics problems, calculating compound interest, measuring elasticity, performing astronomical calculations, assessing reaction rates, and whatnot. This article will cover some of the most common real-life applications of logarithms. The applications are-

1. Measuring the sound intensity

Sounds are measured using a scientific scale called the decibel scale, which works on the principle of logarithms. The scale can measure the faintest whispers and loudest of noises easily and accurately. Further, the decibel scale measures the different intensities of sounds by keeping the normal intensity of the sound as a reference point.

The sound above, below, and on the reference point are all accurately captured by the decibel scale. For instance, 5 times higher than normal intensity sound will be 5db on the decibel scale.

2. Stock market analysis

The ups and downs of the stock market require a lot of calculations and predictions, generally based on highly large values and calculations. The data obtained in the stock market are generally expressed in exponential form. Applying and understanding the exponential form of data in every situation is impossible.

Hence it is converted into logarithms for better understanding. Data transformation and normalization during stock market analysis are also carried out using the multiplicative properties of logarithms.

3. Studying the process of decay of radioactive elements

Another great application of logarithms in real life is studying the exponential decay process of radioactive elements. The decay of radioactive elements varies to a wide range, from a few seconds to decades. Examples of Exponential decay and growth both utilize the use of logarithms.

Logarithms help to study the half-life of radioactive elements and assess the decay rate and other processes related to radioactive elements, such as medical imaging.

4. Radiocarbon dating

Radiocarbon dating is the process of studying the age of geological and archaeological objects infused with radioactive carbon isotopes (C-14) in organic matter. This process helps to identify how old an object is and how long it could have been preserved without decaying and provides an insight into a human’s history timeline. Logarithms here help to convert complex exponential decay equations into simple linear equations for a better understanding of laymen and professionals.

5. Assessing the magnitude of earthquakes using the Richter scale

Richter scale is a scale widely used for assessing the magnitude of earthquakes by studying the disturbances in the plates of the earth’s layers. This scale works on the principle of logarithms. The energy released from the seismic waves is generally very high in intensity, requiring exponential equations to represent it.

Also, comparing the magnitude of two or more two earthquakes requires logarithms to solve the complex exponential equations. Logarithms simplify the representation and understanding of complex and large exponential values, resulting in better application of the obtained information.

6. Measuring pH levels of chemicals

Logarithms are widely applied in the process where studying pH level is important. pH levels help to identify the acidic and alkaline nature of the substances, such as soil, chemical elements, etc. pH levels can be studied through a pH scale that works based on the concept of logarithms.

The PH scale helps determine the concentration of hydrogen ions in the elements or substances and the application of Hydrogen ion-infused elements in various chemical and environmental processes.

7. Calculating the growth of the human species or other living species

Logarithms are also applied in the biological field for assessing and calculating the growth of human species and other living species, generally calculated in exponential form. Logarithms come in handy in studying the exponential growth of populations, the speed of doubling the population, comparing the growth rate of two populations predicting the future growth or decline rate of the population, and aiding in studies related to environment and biodiversity.

Studying human populations using logarithms helps in better planning and making policies for the welfare of the citizens, whereas studying bacteria or other living species populations is useful for understanding their life cycles and processes.

8. Data compression

Ever wondered how such large data obtained from various organizations and governments is stored and used? Storing such large data requires various storage devices, which will still be insufficient. In that case, the data compression technique is used, which uses logarithms to simplify and compress the data.

The data is compressed using different coding processes, such as arithmetic, transform, Huffman, Delta, entropy, Shannon-Fano, run-length encoding, etc., based on logarithm and its application.

9. Analyzing drug concentrations in medicines

Pharmacology and pharmacokinetics also have major applications of logarithms in identifying drug concentration in medicines and regulating the use of drugs. Logarithms are also used to study the change in the concentration of drugs in the human body over time.

Further, logarithms help to assess how long the drug will stay in the human body and when the effects will wear off. Whether the drug is distributed equally in the whole body or not is also studied through logarithms.

10. Sensation and perception

Another important area where logarithms are used is in the scientific study of sensation and perception using the highly renowned Weber’s and Steven’s power laws. Psychophysics and signal detection theory (SDT) also require the application of logarithms to estimate the presence or absence of stimulus and the intensity of the stimulus present. Identifying the magnitude of stimulus and sensory adaptation requires logarithmic scaling.

Concluding thoughts

Logarithms is a highly used mathematical concept that makes the findings, representation, and interpretation of data from various disciplines easy and convenient. Understanding and applying logarithms in real life makes various complex processes easy to understand. Solving logarithmic equations is a discovery learning activity that helps discover and predict various phenomena.

essay on how logarithms used in real life

I am Sehjal Goel, a psychology student, and a writer. I am currently pursuing my Masters’s from Banaras Hindu University, Varanasi. Child psychology has always fascinated me and I have a deep interest in learning about disabilities in children and spreading awareness regarding the same. My other areas of interest are neuropsychology and cognitive psychology. Connect me on Linkedin

Ok, ok, we get it: what are logarithms about?

Logarithms find the cause for an effect, i.e the input for some output

A common "effect" is seeing something grow, like going from \$100 to \$150 in 5 years. How did this happen? We're not sure, but the logarithm finds a possible cause: A continuous return of ln(150/100) / 5 = 8.1% would account for that change. It might not be the actual cause (did all the growth happen in the final year?), but it's a smooth average we can compare to other changes.

By the way, the notion of "cause and effect" is nuanced. Why is 1000 bigger than 100?

100 is 10 which grew by itself for 2 time periods ($10 * 10$)
1000 is 10 which grew by itself for 3 time periods ($10 * 10 * 10$)

We can think of numbers as outputs (1000 is "1000 outputs") and inputs ("How many times does 10 need to grow to make those outputs?"). So,

Or in other words:

Why is this useful?

Logarithms put numbers on a human-friendly scale.

Large numbers break our brains. Millions and trillions are "really big" even though a million seconds is 12 days and a trillion seconds is 30,000 years. It's the difference between an American vacation year and the entirety of human civilization.

The trick to overcoming "huge number blindness" is to write numbers in terms of "inputs" (i.e. their power base 10). This smaller scale (0 to 100) is much easier to grasp:

power of 0 = $10^0$ = 1 (single item)
power of 1 = $10^1$ = 10
power of 3 = $10^3$ = thousand
power of 6 = $10^6$ = million
power of 9 = $10^9$ = billion
power of 12 = $10^12$ = trillion
power of 23 = $10^23$ = number of molecules in a dozen grams of carbon
power of 80 = $10^80$ = number of molecules in the universe

A 0 to 80 scale took us from a single item to the number of things in the universe. Not too shabby.

Logarithms count multiplication as steps

Logarithms describe changes in terms of multiplication: in the examples above, each step is 10x bigger. With the natural log, each step is "e" (2.71828...) times more.

When dealing with a series of multiplications, logarithms help "count" them, just like addition counts for us when effects are added.

Show me the math

Time for the meat: let's see where logarithms show up!

Six-figure salary or 2-digit expense

We're describing numbers in terms of their digits, i.e. how many powers of 10 they have (are they in the tens, hundreds, thousands, ten-thousands, etc.). Adding a digit means "multiplying by 10", i.e.

$\displaystyle{1 \text{[1 digit]} \cdot 10 \cdot 10 \cdot 10 \cdot 10 \cdot 10 \text{[5 more digits]} = 10^5 = 100,000}$

Logarithms count the number of multiplications added on , so starting with 1 (a single digit) we add 5 more digits ($10^5$) and 100,000 get a 6-figure result. Talking about "6" instead of "One hundred thousand" is the essence of logarithms. It gives a rough sense of scale without jumping into details.

Bonus question: How would you describe 500,000? Saying "6 figure" is misleading because 6-figures often implies something closer to 100,000. Would "6.5 figure" work?

Not really. In our heads, 6.5 means "halfway" between 6 and 7 figures, but that's an adder's mindset. With logarithms a ".5" means halfway in terms of multiplication, i.e the square root ($9^.5$ means the square root of 9 -- 3 is halfway in terms of multiplication because it's 1 to 3 and 3 to 9).

Taking log(500,000) we get 5.7, add 1 for the extra digit, and we can say "500,000 is a 6.7 figure number". Try it out here:

We geeks love this phrase. It means roughly "10x difference" but just sounds cooler than "1 digit larger".

In computers, where everything is counted with bits (1 or 0), each bit has a doubling effect (not 10x). So going from 8 to 16 bits is "8 orders of magnitude" or $2^8 = 256$ times larger. ("Larger" in this case refers to the amount of memory that can be addressed.) Going from 16 to 32 bits means an extra 16 orders of magnitude, or $2^16$ ~ 65,536 times more memory that can be addressed.

Interest Rates

How do we figure out growth rates? A country doesn't intend to grow at 8.56% per year. You look at the GDP one year and the GDP the next, and take the logarithm to find the implicit growth rate.

My two favorite interpretations of the natural logarithm (ln(x)), i.e. the natural log of 1.5:

Assuming 100% growth, how long do you need to grow to get to 1.5? (.405, less than half the time period)
Assuming 1 unit of time, how fast do you need to grow to get to 1.5? (40.5% per year, continuously compounded)

Logarithms are how we figure out how fast we're growing.

Measurement Scale: Google PageRank

Google gives every page on the web a score (PageRank) which is a rough measure of authority / importance. This is a logarithmic scale, which in my head means "PageRank counts the number of digits in your score".

So, a site with pagerank 2 ("2 digits") is 10x more popular than a PageRank 1 site. My site is PageRank 5 and CNN has PageRank 9, so there's a difference of 4 orders of magnitude ($10^4$ = 10,000).

Roughly speaking, I get about 7000 visits / day. Using my envelope math, I can guess CNN gets about 7000 * 10,000 = 70 million visits / day. (How'd I do that? In my head, I think $7k * 10k = 70 * k * k = 70 * M$). They might have a few times more than that (100M, 200M) but probably not up to 700M.

Google conveys a lot of information with a very rough scale (1-10).

Measurement Scale: Richter, Decibel, etc.

Sigh. We're at the typical "logarithms in the real world" example: Richter scale and Decibel. The idea is to put events which can vary drastically (earthquakes) on a single scale with a small range (typically 1 to 10). Just like PageRank, each 1-point increase is a 10x improvement in power. The largest human-recorded earthquake was 9.5; the Yucatán Peninsula impact, which likely made the dinosaurs extinct, was 13.

Decibels are similar, though it can be negative. Sounds can go from intensely quiet (pindrop) to extremely loud (airplane) and our brains can process it all. In reality, the sound of an airplane's engine is millions (billions, trillions) of times more powerful than a pindrop, and it's inconvenient to have a scale that goes from 1 to a gazillion. Logs keep everything on a reasonable scale.

Logarithmic Graphs

You'll often see items plotted on a "log scale". In my head, this means one side is counting "number of digits" or "number of multiplications", not the value itself. Again, this helps show wildly varying events on a single scale (going from 1 to 10, not 1 to billions).

Moore's law is a great example: we double the number of transistors every 18 months (image courtesy Wikipedia ).

The neat thing about log-scale graphs is exponential changes (processor speed) appear as a straight line. Growing 10x per year means you're steadily marching up the "digits" scale.

Onward and upward

If a concept is well-known but not well-loved, it means we need to build our intuition. Find the analogies that work, and don't settle for the slop a textbook will trot out. In my head:

Logarithms find the root cause for an effect (see growth, find interest rate)
They help count multiplications or digits, with the bonus of partial counts (500k is a 6.7 digit number)

Happy math.

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How Logarithms are Used in Real Life?

The Origin and Discovery of Logarithms

A logarithm, or log, is a mathematical operation. A logarithm consists of a base; when multiplied by itself a specific number of times, it reaches another number. For example, log 2 (64) equals 6, which means that if you multiply the base 2 six times with itself, it becomes 64. The logarithmic base 2 of 64 is 6. The log base of 10 of 100 equals 2, so you get to 100 by multiplying 10 twice. A logarithm is the inverse of an exponential, that is, 2 6 equals 64, and 10 2 equals 100. Sounds complicated to you? John Napier, a Scottish mathematician, discovered logarithms. To further his interest in astronomy, he needed to expedite tedious and complex astronomical calculations involving large numbers. John worked in an age before the invention of calculators.

Logarithms in Real Life

If you have a goal of making a million dollars in the next decade, the formula that your investment advisor will tell you will most likely be a logarithmic one. You will need to invest x at y% annual interest for a decade to get returns worth a million dollars. While it is difficult to visualize a million dollars quickly in your head, notice how much easier it becomes once you know x and y. Logarithms simplify insights involving large figures, such as the number of visits per day on Google’s search home page, earthquake intensity readings, or sound intensity readings of a commercial airplane during take off. There are numerous applications of logarithms due to their ability to “scale down” large numbers in a human-friendly manner. Even after the invention of calculators and supercomputers, centuries after John Napier’s discovery, logarithms are still in use.

Real-life Uses of Logarithms

Think of a scenario where you need to interpret extremely large numbers and dumb them down. You will need logarithmic operations to get consumable insights. How does it simplify computation for a computer? Binary search algorithm design is an example of how logarithmic operations can help locate any name among a million in a phone directory with just 20 comparisons.

Another modern-day application is in space shuttle launches. The velocity, distance traversed, and curved path of a falling rocket incorporate logarithmic operations.

Log transformation of images involves replacing every pixel of an image with its log. The transformed image appears more enhanced with better quality. Logarithmic image processing (LIP) models are mathematical frameworks used extensively in the field of digital image processing algorithms. LIP models improve characteristics such as contrast and sharpness of an image. The applications of logarithmic image transformation extend to various fields, including medical imaging, satellite imaging, robotic vision, and remote sensing.

How Logarithms Make Our Life Easy?

Logarithms rescue us from having to process both large numbers as well as extremely small numbers. For instance, how can you determine how safe it is to indulge in a risky habit such as smoking, driving a bike without a helmet, or bungee jumping down a valley? There is data available in these scenarios that will read something like this hypothetically. In your country, 1 in 20,000 people died in bike accidents because they weren’t wearing a helmet. How do you use this information? Every year, 1 in 100,000 people dies from bungee jumping. Does that mean it is safer than riding a bike without a helmet? The larger the numbers, the more scrambled the math gets in our brains. Logarithms to the rescue! Think of a safety indicator on a scale of 1 to 10, with 1 being the most unsafe and 10 being the safest activity. If 1 in X people succumbs to their indulgence in an activity, the safety measure of that activity will be log 10 (X). The log 10 (20000) value is 4.3, and the log10(100000) value is 5. This can be interpreted to mean bungee jumping is slightly safer than riding a bike without a helmet!

Measuring a substance’s acidity or alkalinity is easier with logarithms too. Our brains find it difficult to process numbers that are too small! For instance, water contains 1*10 -7 moles of hydrogen ions per liter. So, is it acidic? And how acidic is it compared to washing soap? This is where a standard pH scale comes in handy and rescues you from having to bother with extremely small numbers. The pH scale ranges anywhere between 0 and 14, making water a neutral solution with a pH of 7. Substances such as washing soap with a pH greater than 7 are acidic.

A logarithm is an instinctive concept by nature. Thanks to this, it has widespread applications in areas that require converting extremely large and extremely small numbers into understandable insights.

You can find more articles on logarithms and algorithms on BYJU’S FutureSchool blog. Let us know through your comments if you found this blog insightful.

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Logarithm: The Complete Guide (Theory & Applications)

Algebra, Complex Number, Functions & Operations, General Math

The Ultimate Guide to Logarithm - Properties of Logarithm, Complex Logarithm and More!

For the very vast majority of humans on earth, there is a topic found in the good old math textbooks that many of us still even dread contemplating about, as it seems to mess with our brain in a rather particular way. The name? Logarithmus — or Logarithm in English to be sure!

As terrible-sounding as it is, logarithm seems to have this distinct characteristic of metaphorically leaving a bad taste in our mouth. In fact, even for those who managed to maneuver around it back in high school, logarithm still remains largely as an evasive concept. The “I-can-manipulate-expressions-without-understanding-anything” syndrome runs rampant when it comes to logarithm.

Indeed, here in North America, the grade school curriculum has the propensity of overemphasizing the mechanics at the expense of basic theory , leaving us with the formidable task of filling in the logarithmic knowledge gap, which includes — among others — the theory behind the properties of logarithm , and its intended computational use in handling numbers with an order of magnitude veering towards the extremes.

So with that in mind, if you think that the time might have finally come to tame this monster we call logarithm, then it would be our pleasure to congratulate your timing on this very honorable act. And if you are simply looking to explore further into the rabbit hole, that would be doubly appreciated as well, for regardless of your motivation, the taming / musing is on! 🙂

Table of Contents

Logarithm — A Review

Terminology.

Given a real number $x$, one of the challenges in elementary algebra is to express $x$ as a power of another number $b$ (known as the base ). More specifically, we are interested in finding a number $\Box$ such that:

\begin{equation*} x = b^\Box \end{equation*}

As it turns out, this problem — in the crude form that it currently is at least — needs to be patched up first before any meaningful discussion can take place. For example:

If the base is negative , then its powers need not be necessarily well-defined (e.g., $\displaystyle (-e)^{\frac{1}{2}}$).
If the base is $\displaystyle 1$, then any power of it would be just $1$, in which case, it would be impossible for it to generate any number that’s not $1$. A similar remark applies to the case where the base is equal to $0$.

For these reasons, in the context of power determination , it’s customary to require the base $b$ to be a positive number — that is not equal to $1$. While under this assumption, any power of $b$ would necessarily have to be positive , it would also transpire —under this setup — that any positive number can be expressed as a power of $b$ in a unique way. That is, as long as $x$ is positive , there will be a unique number $\Box$ (known as the exponent ) such that:

\begin{equation*} x= b^{\Box} \end{equation*}

in which case, we will simply call $\Box$ the logarithm of $x$ (in base $b$). In other words, logarithm is basically what happens when we expressed a number as a power , and then take the exponent from that power — It gives us the magnitude of a number, with respect to the base in question.

For example, when we try to express the number $64$ as a power of $2$, we get that $64= 2^6$. This alone shows that $6$ is the logarithm of $64$ — with respect to the base $2$.

Notation-wise, the logarithm of $x$ in base $b$ is denoted by $\log_b x$, with $x$ also being called the argument of the logarithm. When considered as a function, $\log_b x$ is defined on all positive numbers — as long as the base $b$ is valid (i.e., $\displaystyle b>0, b \ne 1$) .

To begin, we first note that regardless of the value of the base $b$, we always have that:

$\displaystyle \log_b 1 = 0$ (since $0$ is the number $b$ needs to be raised to yield $1$)
$\displaystyle \log_b b = 1$ (since $1$ is the number $b$ needs to be raised to yield $b$)
$\displaystyle \log_b \frac{1}{b} = -1$ (since $-1$ is the number $b$ needs to be raised to yield $\displaystyle \frac{1}{b}$)

Because these results are almost immediate and sufficiently notable, we’ll simply refer to them as the trivial logarithmic identities .

In addition, since $\log_b x$ stands for the number which exponentiates to $x$, we also have that by definition:

\begin{align*}b^{\log_b x} & = x \qquad (\text{for all }x>0)\end{align*}

On the other hand, we also have that:

\begin{align*} \log_b (b^x) = x \qquad (\text{for all } x \in \mathbb{R}) \end{align*}

Since one can see by inspection that $x$ is precisely the number which exponentiates to $b^x$.

For example, since $\displaystyle \log_2 53$ is the number that $2$ needs to raise to yield $53$, we have that $\displaystyle 2^{\log_2 53} =53$. Similarly, since $\displaystyle 10^{-\pi}$ is a power of $10$ with the exponent $-\pi$, we can infer that $\displaystyle \log_{10} \left(10^{-\pi}\right) = -\pi$.

Common Logarithm (Base 10)

Being the inverse of the exponential function $\displaystyle 10^x$, the base-$10$ logarithmic function — also known as the common logarithm — is customarily denoted by $\log_{10} x$, $\log x$, or simply $\lg x$ for short. The common logarithm is of great interest to us, primarily due to the prevalence of the decimal number system in various cultures around the world.

Note that in older scientific texts and some textbooks in higher mathematics, $\log x$ can also refer to — and usually is — the natural logarithm of base $e$ .

When the common logarithm of a number is calculated, the decimal representation of the logarithm is usually split into two parts: the integer component (a.k.a., characteristic ) and the fractional component (a.k.a., mantissa ). The characteristic in essence tells us the number of digits the original number has, and the mantissa hints at the extent to which this number is close to its next power of $10$. These are the facts that make common logarithm a particularly handy tool in determining the order of magnitude of an exceptionally large (or small ) number.

For example, to figure out the magnitude of the number $50!$ (i.e., $50 \times \cdots \times 1$), we proceed to calculate its logarithm, yielding that: \[ \log (50!) \approx 64.483 \] which means that $50! \approx 10^{64.483} =$ $10^{64}10^{0.483} \approx$ $10^{64} \cdot 3.04$, suggesting that $50!$ is a $65$ -digit number which starts with $3$ — the characteristic $64$ gives away the number of digits, and the mantissa $0.483$ reveals the rest about the number itself.

Take home message? There is no need to write out a number in full to figure out its approximate size !

Binary Logarithm (Base 2)

Being the inverse of the exponential function $2^x$, the binary logarithm function $\log_2 x$ is extensively used in the field of computer science , primarily due to the fact that computers store information in bits (i.e., digits which takes $0$ or $1$ as possible values).

Similar to the case in base $10$, binary logarithm can be used to figure out the number of digits of a positive integer in binary representation . In addition, binary logarithm is also used to figure out the depth of a binary tree , or even the number of operations required by certain computer algorithms (this falls into a topic known as algorithmic time complexity ).

Beyond the world of computers, binary logarithm is also used in music theory to conceptualize the highness of musical notes, based on the fundamental observation that raising a note by an octave increases the frequency of the note by twofold . As a result, it is often convenient to conceive a musical interval as the binary logarithm of the frequency ratio .

Natural Logarithm (Base $e$)

In some textbooks concerned with a more rigorous development of transcendental functions , the base-$\displaystyle e$ logarithmic function — otherwise known as natural logarithm , $\log_e x$ or simply $\ln x$ — are sometimes defined as the area between the reciprocal function $\frac{1}{x}$ and the x-axis from $1$ to $x$ (hence the term natural ).

Natural Logarithm and the Divergence of the Harmonic Series

Under this definition, it could be shown that the inverse of $\ln x$ is precisely the natural exponential function $e^x$, leading to the following standard definition of natural logarithm:

Given a positive number $x$, $\ln x$ denotes the number $e$ needs to be raised — in order to become $x$.

Unlike the number $10$ — which is preferred due to the prevalence of decimal numbering system — the number $\displaystyle e$ is one of the special constants that pops up surprisingly often in various mathematical discourses — irrespective of the number system being chosen. As a result, mathematicians tend to consider base $e$ as more natural than base $10$ — even though some applied scientists and engineers beg to differ in various occasions…

Actually, to illustrate the scope of these intellectual biases among the scientific community, here’s an interesting account from Wikipedia on the historical development of the notations for logarithms:

Because base 10 logarithms were most useful for computations, engineers generally simply wrote “log(x)” when they meant log 10 (x). Mathematicians, on the other hand, wrote “log(x)” when they meant log e (x) for the natural logarithm. Today, both notations are found.

Since hand-held electronic calculators are designed by engineers rather than mathematicians, it became customary that they follow engineers’ notation. So the notation, according to which one writes “ln(x)” when the natural logarithm is intended, may have been further popularized by the very invention that made the use of “common logarithms” far less common, electronic calculators.

Logarithm of an Arbitrary Base

Graphs of the Logarithmic Functions of base 2, e and 10

In addition to the three most popular logarithmic functions introduced earlier, one can also define logarithm using other valid bases as well. In practice, logarithm is generally employed with the intention of condensing large numbers (i.e., greater than $1$) into smaller numbers, so that the larger the base, the smaller the logarithm.

However, that’s only part of the story, as all the logarithmic functions we have encountered so far have bases exceeding the number $1$ (i.e., large base ). In fact, in the cases where the base is strictly between $0$ and $1$ (i.e., small base ), the graph of the logarithmic function will be turned upside down . Indeed, as opposed to a standard logarithmic function which increases from $-\infty$ to $\infty$, a small-base logarithmic function actually decreases from $+\infty$ to $-\infty$ as the argument increases.

Fortunately though, we rarely have to resort to this kind of logarithmic function in practice. As we shall see later with the Change of Base Rule , every pair of logarithmic functions are all but a multiple apart, so that in terms of applications and equation/inequality solving , the three standard logarithms are.generally more than enough to get things going.

Logarithmic Scale and its Applications

Since logarithm allows for mapping an exponential scale into a linear scale , it became an vital concept when it comes to communicating about a numerical variable whose quantities either grow exponentially, or shrink exponentially. Why? Because if we just take the logarithm of that variable, we are in effect turning that variable into something people refer to as a logarithmic scale .

While seemingly a highly-theoretical concept, logarithmic scale — when adopted appropriately — can be used to help us better explain/understand a surprising amount of phenomena found in nature . ranging from stuffs such as the loudness of a sound , the magnitude of an earthquake to the acidity of a solution and the highness of a musical pitch :

In an equal-tempered piano , each key in the piano can be conceived as the binary logarithm of its relative sound frequency, so that every time we press a higher key on the piano, we are in effect increasing the sound frequency by a fixed factor ($\displaystyle \sqrt[12]{2}$ to be precise).
In chemistry , the acidity of a solution is measured in pH , which is defined as the negative logarithm of the concentration of hydrogen ions . This basically means that as pH increases by $1$, the concentration of hydrogen ions decreases tenfold , leading to a substantially less acidic solution in return.
In seismology , the severity of an earthquake can be quantified using the Richter scale , which is essentially the logarithm of the amplitude of seismic waves (relative to a threshold amplitude ), so that every time the number on Richter scale increases by $1$, severity of the earthquake increases tenfold .
In acoustics , the loudness of a sound is generally quantified using decibel (dB), which is a tenth of a bel (B), the latter of which is the logarithm of the sound power (relative to the threshold of hearing ). In practice, this means that every increase of 10 dB (equiv., 1B), increases the sound power of a source by tenfold .

Decibel - The Different Levels of Loudness in Logarithms

While not all logarithmic scales share the same base, the fact that one can hijack a concept such as logarithm — originally a purely-computational device — into an unifying abstract framework stringing together various seemingly unrelated phenomena found in nature, illustrates the power of the human mind and by extension, the thin line between mathematical invention and discovery .

Properties of Logarithm

Properties involving the arguments.

One of the reasons why logarithm was such a powerful computational tool back in the old days — before the invention of computers or calculators — lies in the fact that one can always leverage certain properties of logarithm to reduce a complicated argument to its individual constituents — and doing so irrespective of the base in question. In what follows, we lay out five of such properties, which pertains to the product , the reciprocal , the quotient , the power and the root of a logarithm, respectively.

Product Rule

Given a product $xy$ and a base $b$, can we find the logarithm of $xy$ in terms of the logarithms of $x$ and $y$? As it turns out, the answer is a resounding yes , and a bit of inspection shows that $\log x + \log y$ is the number we are looking for. How so? Because that’s the number $b$ needs to be raised to get to $xy$:

\begin{align*} b^{\log x + \log y}= b^{\log x} \cdot b^{\log y} = xy \end{align*}

This fact — that logarithm of a product can be reduced into sum of logarithms of its constituents — gives rise to a property commonly known as the Product Rule .

Rule 1 — Product Rule for Logarithm

Given any two positive numbers $x$, $y$, we have that:

\begin{align*} \log (xy) =\log x + \log y \end{align*}

where all logarithms are assumed to be under the same valid base $b$.

In particular, when the base is $10$, the Product Rule can be translated into the following statement:

The magnitude of a product, is equal to the sum of its individual magnitudes.

For example, to gauge the approximate size of numbers like $365435 \cdot 43223$, we could take the common logarithm, and then apply the Product Rule , yielding that:

\begin{align*} \log (365435 \cdot 43223) & = \log 365435 + \log 43223 \\ & \approx 5.56 + 4.63 \\ & = 10.19 \end{align*}

which shows that $\displaystyle 365435 \cdot 43223$ is a 11-digit number close to $\displaystyle 10^{10.19} \approx 1.55 (10^{10})$.

Only apply the Product Rule when the preconditions are met. For example, one thing we cannot do is to break down $9$ into $-1$ and $-9$, and claim that $\ln 9 = \ln(-1) + \ln (-9)$.

And because an equality is by default bidirectional , instead of breaking the product by using the Product Rule from the left to the right , we can also use it from the right to the left , thereby turning a sum of logarithms into a product instead. For example:

\begin{align*} \log 25 + \log 4 = \log (25 \cdot 4) = \log 100 = 2 \end{align*}

On the downside however, since logarithm by default takes only positive numbers as arguments, applying logarithm and its properties to a function or an equation can significantly restrict its domain of feasibility . For example, while the function $x^2$ is defined on all real numbers, once we take the logarithm and apply the Product Rule , the resulting equality is still applicable — but now to the positive numbers only :

\begin{align*} \log (x^2) = \log x + \log x = 2 \log x\end{align*}

which serves as a good reminder that any logarithm-based algebraic technique — be it logarithmic equation solving , logarithmic inequality solving or logarithmic differentiation — should be carried out with this potential restriction in mind.

Reciprocal Rule

We know that every positive number has a multiplicative inverse (i.e., reciprocal ), so perhaps there is also a shortcut in finding the logarithm of a reciprocal? Here again, the answer is a resounding yes . To see why, suppose that we are are given a positive number $x$, then by Product Rule , we have that:

\begin{align*} \log \left(x \ \cdot \frac{1}{x} \right) & = \log x + \log \left( \frac{1}{x} \right) \end{align*}

On the other hand, we also have that:

\begin{align*} \log \left(x \cdot \frac{1}{x} \right) & = \log 1 = 0 \end{align*}

Bridging the two equations together, we get that:

\begin{align*} \log x + \log \left( \frac{1}{x} \right) & = 0 \end{align*}

Or equivalently,

\begin{align*} \log \left( \frac{1}{x} \right) & = – \log x \end{align*}

Surprise! We just discovered the Reciprocal Rule , which states that to find the logarithm of a reciprocal, we just have to negate the logarithm of the original number.

Rule 2 — Reciprocal Rule for Logarithm

Given any positive number $x$, we have that:

\begin{align*} \log \left(\frac{1}{x} \right) = -\log x \end{align*}

where all logarithms are assumed to be under the same valid base $b$.

So instead of calculating the binary logarithm of $\displaystyle \frac{1}{512}$ from scratch, we could turn its head around and do:

\begin{align*} \log_2 \left( \frac{1}{512} \right) = -\log_2 512 = -9 \end{align*}

Quotient Rule

Now that both Product Rule and Reciprocal Rule are in order, let’s see what happens if we apply them to a quotient of positive numbers $x$ and $y$:

\begin{align*} \log \left( \frac{x}{y} \right) & = \log \left( x \cdot \frac{1}{y} \right) \\ & = \log x + \log \left( \frac{1}{y} \right) \\ & = \log x-\log y \end{align*}

Bingo! We have just shown that the logarithm of a quotient is precisely the difference between the original logarithms — a property commonly known as the Quotient Rule .

Rule 3 — Quotient Rule for Logarithm

Given any two positive numbers $x$ and $y$, we have that:

\begin{align*} \log \left( \frac{x}{y} \right) =\log x-\log y \end{align*}

For example, instead to computing the natural logarithm of $\displaystyle \frac{2}{e}$ from scratch, we could apply the Quotient Rule , and get that:

\begin{align*} \ln \left( \frac{2}{e}\right) = \ln 2-\ln e = \ln 2-1 \end{align*}

As in the case with Power Rule , instead of breaking the quotient , we can also use the Quotient Rule from the right to the left , thereby turning a difference into a quotient instead. For example:

\begin{align*} \log 45-\log 9 = \log \left( \frac{45}{9} \right) = \log 5 \end{align*}

In base $10$, the Quotient Rule can also be translated into the following insight:

The magnitude of a quotient, is equal to the difference of the individual magnitudes.

which explains why in natural science, a quantity is often expressed in logarithmic scale , by taking the logarithm of the ratio between the said quantity and a reference point .

As for the logarithm of a number raised to an integer power, we begin by noting the case where a number is raised to $0$:

\begin{align*} \log (x^0) = \log 1 = 0 = 0 \log x \end{align*}

In the case where a number is raised to a positive integer $n$, the logarithm can be obtained through the repeated applications of Product Rule :

\begin{align*} \log (x^n) & = \log \underbrace{ \left(x \cdots x \right)}_{n \text{ times}} \\ & = \underbrace{\log x + \, \dots + \log x}_{n \text{ times}} \\ & = n \log x \end{align*}

And in the case where a number is raised to a negative integer of the form $-n$, a mix of Product Rule and Reciprocal Rule will do:

\begin{align*} \log (x^{-n}) & = \log \left[ \left( \frac{1}{x}\right)^n \right] \\ & = n \log \left( \frac{1}{x}\right) \\ & = -n \log x \end{align*}

Either way, we’ve just shown that when a number is raised to a integer power, the resulting logarithm is rescaled precisely by that power as well. This interesting finding would result in a key property of logarithm known as the Power Rule .

Rule 4 — Power Rule for Logarithm

Given any positive number $x$ and integer $n$, we have that:

\begin{align*} \log (x^n) = n \log x \end{align*}

As some might have expected, the Power Rule is by itself a very powerful property. For one, it allows us to pull out the exponent from the argument of a logarithm, thereby normalizing a potentially gigantic / minuscule number (e.g., $\displaystyle \log_2 (3^{15}) = 15 \log_2 3$), Conversely, the Power Rule can also be used to push an exponent inside the argument of a logarithm, thereby producing a potentially-simpler expression (e.g., $\displaystyle 3 \ln 5 = \ln (5^3) = \ln 125$).

Make sure that the precondition is met before applying the Power Rule. For example, while $\ln (x^8)$ is defined on all non-zero numbers, the equation $\ln (x^8)=8 \ln x$ is only true when $x>0$. In this case though, the issue can be resolved by absolutizing $x$, yielding the equality $\ln (|x|^8) = 8 \ln |x|$ instead.

Similar to the case with Product Rule and Quotient Rule , Power Rule can be interpreted as follows in base $10$:

To find some shortcut in evaluating the logarithm of a root , we begin by observing that for all positive integer $n$, an application of Power Rule shows that:

\begin{align*} \log \left[ (\sqrt[n]{x})^n \right] = n \log (\sqrt[n]{x}) \end{align*}

\begin{align*} \log \left[ (\sqrt[n]{x})^n \right] = \log x \end{align*}

Bridging the two equalities together, we get that:

\begin{align*} n \log (\sqrt[n]{x}) = \log x \end{align*}

\begin{align*} \log (\sqrt[n]{x}) = \frac{\log x}{n} \end{align*}

Awesome! This shows that to figure out the logarithm of a $n$ th root, all we have to do is to divide the logarithm of the original number by $n$ — An insight which results in another property of logarithm known as the Root Rule :

Rule 5 — Root Rule for Logarithm

Given any positive number $x$ and positive integer $n$, we have that:

\begin{align*} \log (\sqrt[n]{x}) = \frac{\log x}{n} \end{align*}

Much like the Power Rule , the Root Rule is not only useful for its ability to pull out the root from the logarithm (as in $\displaystyle \log (\sqrt[12]{6}) = \frac{\log 6}{12}$), but for its ability to create a root out of nothing as well (as in $\displaystyle \frac{\ln 2}{5}=\ln (\sqrt[5]{2})$).

In base $10$, the Root Rule can be interpreted as follows:

When a number is rooted, the resulting magnitude is rescaled precisely by the degree of the root in question.

And when we combine Power Rule and Root Rule together, we get that for any rational number of the form $\displaystyle \frac{m}{n}$ ($m \in \mathbb{Z}, n \in \mathbb{N}$):

\begin{align*} \log x^{\frac{m}{n}} & = \log \left[ (\sqrt[n]{x})^m \right] \\ & = m \log (\sqrt[n] x) \\ & = \frac{m}{n} \log x \end{align*}

In fact, this is nothing more than a special instance of the Generalized Power Rule :

\begin{align*} \log (x^p) = p \log x \qquad (\text{for all }p \in \mathbb{R}) \end{align*}

which can be proved by showing that $p \log x$ is indeed the exponent to which the base $b$ needs to be raised — to produce $x^p$:

\begin{align*} b^{p \log x} & = \left( b^{\log x }\right)^p = x^p\end{align*}

And finally, here is an example illustrating all the argument-related properties of logarithm we have seen thus far:

\begin{align*} \log \left( \frac{5^3 \cdot \sqrt[4]{15}} {10^{66} \cdot e^{\pi}} \right) & = \log \left( 5^3 \cdot \sqrt[4]{15} \right) – \log \left( 10^{66} \cdot e^{\pi} \right) \\ & = \left [\log 5^3 + \log \sqrt[4]{15} \right] – \left[ \log (10^{66}) + \log (e^{\pi})\right] \\ & = 3 \log 5 + \frac{\log 15}{4} – 66 \log 10 – \pi \log e \end{align*}

Properties Involving the Bases

OK. So that was a bit of properties involving the argument of logarithm, but what if we want to tweak with the base instead? Well, we’ve got you covered on that one too! In what follows, we present four properties of logarithm involving the bases for your own pleasure. These are the Chain Rule , the Change-of-Base Rule , the Base-Swapping Rule and the Base-Argument Interchangeability .

Suppose for a moment that you have mastered all the five properties of logarithm used to reduce a argument into its simplest constituents, but for one reason or another find the base rather annoying. What would you do?

In our case, we would to find an alternate formula for calculating the same logarithm — without having to resort to this base directly .

Actually, let’s begin by tightening the question a bit: given a positive number $x$, is there a way of calculating $\log x$ (under base $b$) using a new base $a$? Or even better: what’s an expression involving $a$, such that when $b$ raised to it, becomes $x$?

Here, to find one such expression, it would be just natural to start with $\displaystyle \log_a x$ and see where it takes us from there. By definition, $\displaystyle \log_a x$ is just the number that $a$ needs to be raised, to become $x$. That is:

\begin{align*} a^{\log_a x} = x \end{align*}

But herein lies a problem: we want the left hand side to be a power of $b$ though. How can we turn that $a$ into $b$? Well, if we write $a$ as a power of $b$ that is! With this newfound insight, we proceed to replace the bottom $a$ with $\displaystyle b^{\log a}$, yielding that:

\begin{align*} {\left( b^{\log a} \right)}^{\log_a x} = b^{\log a \cdot \log_a x} = x \end{align*}

Bingo! The exponent in the middle term is exactly what we were looking for — the expression that $b$ needs to be raised, to become $x$! We can therefore conclude that:

\begin{align*} \log x = \log a \cdot \log_a x \end{align*}

In English, this reads:

The logarithm of a number, can be evaluated as the logarithm of a new base , times the logarithm of the original number under that new base.

In fact, this property is so impressive, that we decided to baptize it as the Chain Rule (not to be confused with the usual chain rule in calculus ).

Rule 6 — Chain Rule for Logarithm

Given any positive number $x$ and valid base $a$, we have that:

where all the logarithms whose base isn’t explicitly defined are assumed to be under the same valid base $b$.

Here, an example illustrating its use is definitely called for: suppose that we’re given the task of determining the magnitude of the number $1024$ (i.e., find $\log_{10} 1024$), but figure that it would easier if the base were in $2$ instead, then one thing that we can do would be to apply the Chain Rule with $2$ as the new base , yielding that:

\begin{align*} \log_{10} 1024 = \log_{10} 2 \cdot \log_2 1024 = \log_{10} 2 \cdot 10 \approx 3.01 \end{align*}

so the magnitude of $1024$ is approximately $3.01$ (i.e., $1024 = 10^{3.01}$), which is consistent with it being a 4-digit number.

In fact, here is more: it actually doesn’t matter which new base we choose to use! The base could have been $3$, $e$ or even $\pi$, and the result would have been exactly the same!

\begin{align*} \log_{10} 1024 & = \log_{10} 3 \cdot \log_3 1024 \\ & = \log_{10} e \cdot \log_e 1024 \\ & = \log_{10} \pi \cdot \log_{\pi} 1024 \end{align*}

Change-of-Base Rule

If you’re still hanging around, you remember that Chain Rule states that:

Here, if we just solve for $\log_a x$, we would get:

\begin{align*} \log_a x = \frac{\log x}{\log a} \end{align*}

Goodness! Another alternate formula for logarithm! Except that this time, it’s an (in)famous one for real. The name? Change-of-Base Rule !

Rule 7— Change-of-Base Rule for Logarithm

where the logarithms whose base isn’t explicitly defined are assumed to be under the same valid base $b$

At the first sight, this might seem like a mere reformulation of the Chain Rule . However, upon further inspection, one can see that this is actually not the case: what the Chain Rule does is to turn a logarithm into a product of logarithms with different bases, while the Change-of-Base Rule turns a logarithm into a quotient of logarithms with the same base. In addition, the Chain Rule marginally facilitates the evaluation of a logarithm under a new base, while the Change-of-Base Rule actually fully eliminates the dependence on the old base.

In practice, the Change-of-Base Rule is primarily used to compute a “non-standard” logarithm by turning it into a quotient of “standard logarithms” (e.g., $\log_{15} 26 = \frac{\ln 26}{\ln 15} = \frac{\log_2 26}{\log_2 15}$). However, it can also be used in the reverse manner, thereby merging a quotient of logarithms into a single logarithm (e.g., $\frac{\ln 8}{\ln 2} = \log_2 8 = 3$).

In fact, both the Chain Rule and the Change-of-Base Rule provide a unifying framework for logarithms of all valid bases, by showing that every logarithmic function is a multiple apart from one another. With the advent of computers and calculators , the Change-of-Base Rule also opens up the practice of calculating a logarithm of an arbitrary base, by standardising it to base $e$, or — if a scientific calculator is used — to base $10$.

Base-Swapping Rule

Now, let’s do something fun. Remember that the Change-of-Base Rule states that:

\begin{align*}\log_a x = \frac{\log x}{\log a} \end{align*}

with the understanding that the “baseless” logarithms are actually assumed to be under some valid base $b$. Out of curiosity, if we just let this base to be $a$, we get that:

\begin{align*}\log_a x = \frac{\log_a x}{\log_a a} = \log_a x \end{align*}

which is not terribly interesting, as we are basically going in full circle. However, in the very special case where $x \ne 1$, we can let $x$ to be the base instead, thereby producing the following identity:

\begin{align*}\log_a x = \frac{\log_x x}{\log_x a} = \frac{1}{\log_x a}\end{align*}

Impressive! It’s almost like playing LEGO® ! And since no one has given it a name yet, let’s just jump in and baptize it as the Base-Swapping Rule .

Rule 8 — Base-Swapping Rule for Logarithm

Given any two valid bases $x$ and $a$, we have that:

\begin{align*}\log_a x = \frac{1}{\log_x a}\end{align*}

In English, the Base-Swapping Rule translates into the following insight:

An alternate way of figuring out the logarithm of a number under a base, is to find the logarithm of the base under that number instead, and then take the reciprocal.

Terrible pun we know, but if you really hate operating under a certain base, the Base-Swapping Rule provides a quick-and-dirty way to get rid of it. For example:

\begin{align*} \log_{512} 2 = \frac{1}{\log_2 512} = \frac{1}{9}\end{align*}

which shows that mental LEGO can be just as fun as playing with a few dozen pieces of colored plastic . 🙂

Base-Argument Interchangeability

As a general rule of thumb, we don’t want to mess around with the bases and arguments by swapping them around. However, in the very special case where a base is raised to a logarithm , a bit of swapping actually preserves the equality, and is sometimes even preferred — like this:

\begin{align*} x^{\log y} & = y^{\log x} \end{align*}

In case you’re wondering about the exact nature of this black magic, here’s a proof showing how the left-hand side becomes the right-hand side:

( Note : all logarithms are assumed to be under base $b$)

\begin{align*} x^{\log y} & = {\left( b^{\log x} \right)}^{\log y} \\ & = {\left( b^{\log y} \right)}^{\log x} \\ & = y^{\log x} \end{align*}

Impressive property! For the lack of better term, let’s just refer to it as the Base-Argument Interchangeability !

Rule 9 — Base-Argument Interchangeability for Exponent

Given any two positive numbers $x$ and $y$, we have that:

To illustrate the mechanics of this amazing property, here’s a fancy example for your pleasure:

\begin{align*} {\left( 2^{\ln 3} \right)}^{\ln 4} & = {\left( 3^{\ln 2} \right) }^{\ln 4} & (\text{swapping }2 \text{ and }3)\\ & = 4^{\ln \left( 3^{\ln 2} \right) } & (\text{swapping } 3^{\ln 2} \text{ and } 4)\\ & = 4^{\ln \left( 2^{\ln 3} \right) } & (\text{swapping }3 \text{ and }2)\\ & = {\left( 2^{\ln 3} \right)}^{\ln 4} & (\text{swapping }4 \text{ and }2^{\ln 3})\end{align*}

Kind of fun, right? Great place to pull the curtain on the properties of logarithm too! 🙂

Logarithm for Complex Numbers (Optional)

As you might have heard people saying on multiple occasions, logarithm — for most practical purposes at least — is defined only on the positive real numbers. Why? Because a number can only have logarithm if it’s expressible as a power , which in turn must be positive — by virtue of the definition of real-valued exponential functions .

However, as one would expect, this doesn’t sit well with a certain group of mathematical freedom fighters , for whom the following question might be more relevant:

Is there anything we can do to expand the domain of logarithmic functions to other numbers as well?

The answer? Yes, but not without some string attached. As it turns out, forcing this domain expansion will inevitably incur some painful sacrifices on many fronts, which include — among others — a substantial loss in the properties of exponent and logarithm .

Redefining the Exponential Function (of Base $e$)

To begin, we do know that defining the natural exponential function solely on the real numbers is a major stumbling block which needs to go, for it is precisely what restricted the exponentials to the positive numbers in the first place. To extend the definition of the natural exponential function to any complex number , we begin by redefining the exponential function as follows:

\begin{align*} e^{x+yi} \, \stackrel{def}{=} \, e^x \cdot e^{yi} \qquad ( \text{for all }x, y \in \mathbb{R})\end{align*}

which is of course consistent with the original definition of $\displaystyle e^x$ on the real numbers , as it can be seen that for all $\displaystyle x \in \mathbb{R}$:

\begin{align*} e^{x} = e^{x+0i} = e^x \cdot e^{0i} = e^x \cdot e^0 = e^x \end{align*}

But then, how do we interpret this redefinition now that we are in the realm of complex numbers? For one, by Euler’s Formula , we know that $e^{yi}$ stands for the unit complex number with angle $y$. In addition, since multiplying a complex number by a real positive constant results in the rescaling of the number by that same constant, it becomes apparent that the natural exponential function — as defined above — maps a complex number $x+yi$ to another complex number whose length is $e^x$ and whose angle is $y$.

(In fact, this key redefinition of exponential would later play a central role in the development of complex analysis , and in ancillary fields such as Laplace transform as well.)

For example, $\displaystyle e^0=1, e^1=e, e^{\frac{\pi}{2}i}=i, e^{\pi i}=-1$ and $\displaystyle e^{1+ \pi i} =$ $\displaystyle e^1 \cdot e^{\pi i} = -e$. In fact, one can also see that by construction , any exponential is necessarily a number with non-zero length (i.e., an exponential is always non-zero ).

Redefining the Logarithmic Function (of Base $e$)

So the natural exponential function maps the entire complex plane to non-zero complex numbers, but perhaps what is more subtle is the fact that any non-zero complex number can be expressed as a power of $e$ as well. To see why, suppose that we are given a non-zero complex number $\displaystyle z$, then as long as we let:

$x$ be the number such that $e^x$ is equal to the length of $\displaystyle z$ (remember, this is always possible, since $z$ is a non-zero number, and hence must have non-zero length as well)
$y$ be an angle of $\displaystyle z$

then we will have that:

\begin{align*} e^{x+yi} & = e^x \cdot e^{yi} \\ & = \text{the complex number with the length and angle of }z \\ & = z \end{align*}

More specifically, given any non-zero complex number $\displaystyle z$, if $\displaystyle |z|$ stands for the length of $\displaystyle z$ and $\theta$ an angle of $\displaystyle z$, then we have just shown that:

\begin{align*} e^{\ln |z|+\theta i} & = e^{\ln |z|} \cdot e^{\theta i} \\ & = \text{the complex number with the length and angle of } z \\ & = z \end{align*}

In other words, we have just found a logarithm of $z$ — $\displaystyle \ln |z|+\theta i$ that is! However, notice the use of “ a ” instead of “ the “, for as alluded to a bit earlier, this is not going to go all smoothly…

For one, since a complex number can have several equivalent angles , defining the natural exponential function on the entire complex plane can open up a whole can of worms — That is, a whole can of infinitely-many complex numbers whose exponentials are all but the same. For example:

\begin{align*} e^{\pi i}= e^{3 \pi i} = e^{5 \pi i} = e^{- \pi i} = \dots = -1 \end{align*}

which shows that the logarithm of $-1$ — or any other number for that matter — is ill-defined with this current setup.

Naturally, one way to remedy this multivalue-ness of logarithm is to restrict the domain of the exponential function to the principal branch . That is, the set of complex numbers whose imaginary part lies in the interval $\displaystyle (-\pi, \pi]$. With the domain restricted this way, we will then be able to prove that the exponentials of distinct numbers are themselves distinct, making the natural exponential an invertible function — mapping the principal branch to the set of non-zero complex numbers.

Under this setup, the inverse of the exponential function — the natural logarithmic function — maps the set of non-zero complex numbers to the principal branch. In fact, it can be shown that: \[ \boxed{ \ln z = \ln |z| + (\arg{z}) i \ } \]

where $\arg{z}$ stands for the principal angle of $\displaystyle z$ (i.e., the angle in the interval $\displaystyle (-\pi, \pi]$). For example:

\begin{align*} \ln (-1) = 0+\pi i \qquad \ln (-5) = \ln 5 + \pi i \qquad \ln (-1+i)= \ln (\sqrt{2}) + \frac{3\pi}{4}i \end{align*}

Redefining Exponential Functions (of Arbitrary Bases)

With the definition of natural logarithm now taken care of, we can proceed to define the exponential function of an arbitrary base $\displaystyle a$ — simply by standardizing it into base $\displaystyle e$:

\begin{align*} a^{z} = {\left(e^{\ln a}\right)}^{z} \, \stackrel{def}{=} \, e^{(\ln a)z} \end{align*}

Of course, in order for this definition to make sense, the base $\displaystyle a$ has to be non-zero . In addition, the base must not be $\displaystyle 1$ either, otherwise we will just end up with a constant function instead. Basically, this just means that the concept of valid base also has to be updated — from a positive real number that’s not $\displaystyle 1$, to a non-zero complex number that’s not $\displaystyle 1$. And that’s a great achievement if you think about it: the set of valid bases have just gone from the right side of a real number line , to almost the entire complex plane !

Properties of Logarithm — An Update

So all seems to be good. Or is it? Underneath the surface, the properties of logarithm are actually falling apart:

Product Rule fails: $\displaystyle \ln (-1 \cdot -1) = \ln 1 =0$, but $\displaystyle \ln (-1) + \ln (-1) = 2 \pi i$.
Reciprocal/Quotient Rule fails: $\displaystyle \ln \left( 1/(-1) \right) = \ln (-1) = \pi i$, but $\displaystyle \ln 1-\ln(-1) = 0-\pi i = -\pi i$.
Power Rule fails: $\displaystyle \ln \left[ (-1)^2 \right] = \ln 1 = 0$, but $\displaystyle 2 \ln (-1) = 2 \pi i$.
Root Rule fails: The Fundamental Theorem of Algebra implies that a number can now have up to $n$ $n$ th complex roots, so the concept of unique $n$ th root is no longer applicable.

On a brighter side, logarithmic functions of an arbitrary base can now be defined — in terms of $\displaystyle \ln z$ — for all non-zero complex number, using an instance of what’s previously known as the Change-of-Base Rule :

\begin{align*} \log_a x \, \stackrel{def}{=} \, \frac{\ln x}{\ln a} \qquad (x \ne 0, a \text{ being a valid base}) \end{align*}

In light of this, it’s therefore no surprise that the full-fledged Change-of-Base Rule actually hold for complex logarithms in general:

\begin{align*} \log_a x & = \frac{\ln x}{\ln a} \\ & = \frac{\frac{\ln x}{\ln b}}{\frac{\ln a}{\ln b}} \\ & = \frac{\log_b x}{\log_b a} \qquad (x \ne 0, a \text{ and } b\text{ being both valid bases})\end{align*}

In particular, in the case where $x$ is also a valid base , we get that:

\begin{align*} \log_a x = \frac{\log_x x}{\log_x a} = \frac{1}{\log_x a} \end{align*}

What’s the name again? Base-Swapping Rule of course!

In addition, if we just start from the Change-of-Base Rule and solve for $\displaystyle \log_b x$, we get that the Chain Rule is here to stay as well:

\begin{align*} \log_b x = \log_b a \cdot \log_a x \qquad (x \ne 0, a \text{ and } b\text{ being both valid bases}) \end{align*}

In fact, it turns out that even the Base-Argument Interchangeability carries on as well, but only because of a special instance of “ Power Rule ” that we’ve built in into our definition of general exponential function:

\begin{align*} x^{\log_b y} & = e^{\ln x \cdot \log_b y} \\ & = e^{\frac{\ln x \cdot \ln y}{\ln b}} \\ & = e^{\ln y \cdot \log_b x} \\& =y^{\log_b x} \qquad (x, y \ne 0, b \text{ being a valid base}) \end{align*}

So that even though all properties of logarithm involving the arguments are lost in the process, those involving the bases all managed to come out of this unscathed.

Properties of Exponent — An Update

What about the properties of exponents though? In two words: not good . 🙁

Trivial Identities holds: $\displaystyle a^0= 1, a^1=a$.
Additive Properties holds: For all $\displaystyle z_1, z_2 \in \mathbb{C}$, $\displaystyle a^{z_1+ z_2} = a^{z_1} a^{z_2}$ and $\displaystyle a^{z_1 – z_2} = \frac{a^{z_1}}{a^{z_2}}$.
Common-Exponent Properties fails: Given $a,b \ne 0$, we have that $\displaystyle (ab)^x \ne a^x b^x$ and $\displaystyle \left(\frac{a}{b}\right)^x \ne \frac{a^x}{b^x}$ in general.
Power Property fails: $\displaystyle \left( a^{z_1}\right)^{z_2} = a^{z_1 \cdot z_2}$ when the outer exponent $z_2$ is an integer (thanks to the Additive Properties of exponent!), but is otherwise false in general.

Basically, every property — whose validity depends on the argument-related properties of logarithm — will fall out big time.

And finally, to ease the information overload , here’s a summary of the overall gains and casualties associated with expanding the domains of exponential and logarithmic functions :

As long as we restrict to the domain of $\displaystyle e^z$ to a branch , we can define logarithm on all non-zero complex numbers — and doing so using any valid base under the sun.
However, the five properties of logarithm involving the arguments (e.g., Product Rule , Reciprocal Rule , Quotient Rule , Power Rule , Root Rule ) will be lost in the process.
Surprisingly though, the four properties of logarithm involving the bases (i.e., Change-of-Base Rule , Chain Rule , Base-Swapping Rule , Base-Argument Interchangeability ) will be all preserved.
Due to the substantial loss in the properties of logarithm, most of the properties of exponent will be falling apart as well.

So in retrospect, is it worth the effort going through the hurdle of expanding the domain of logarithmic functions? Heck, guess we’ll let you decide on that one! 🙂

Whew! Who would have thought that a pure venture into some basic theory can take us this far! Originally a tool for computing large numbers (e.g., turning products or quotients into sums or differences) and for solving exponential equations (through the properties of logarithm ), logarithm has obviously gone a long way into finding itself in various branches of both applied sciences and pure mathematics .

For the applied scientists, logarithm tends to bring to mind topics such as order-of-magnitude computation (decimal or binary), logarithmic scales (e.g., frequency in musical notes , Richter scale , pH , sound loudness ), algorithmic complexities and log-normal distribution . For others who live in the ivory tower (who’s that?), logarithm is often associated with idealized objects such as the harmonic series , reciprocal funcion , and even prime numbers ,

In fact, when we toyed around the idea of incorporating complex numbers into logarithm, we found that not only were we able to extend the definition of logarithm to the negative real numbers , but to any other non-zero complex number as well. While the extension doesn’t always end up as the way we wanted, doing so actually paves the way of introducing complex logarithm into other seemingly-unrelated subjects, such as integration by partial fractions in calculus for instance.

And before someone’s brain blows up , here’s an interactive table for what we’ve found thus far:

Preliminaries
Properties of (Real) Logarithm
Complex Logarithm

Natural Logarithm (Base e)

Logarithm (arbitrary base), applications.

Assuming that all logarithms below are defined under a valid base $b$, then the following properties hold:

Given two valid bases $a, b$, and that all logarithms with no explicit base are defined under the base $b$, then the following properties hold:

All right. Here’s a wrap for the fascinating topic known as the logarithm, so whether you prefer it real or complex, or you’d rather have it algebraic or applied, at least there’s now no more excuse to avoid them!

About the author

Math Vault and its Redditbots enjoy advocating for mathematical experience through digital publishing and the uncanny use of technologies. Check out their 10-principle learning manifesto so that you can be transformed into a fuller mathematical being too.

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Exponent Rule for Derivative: Theory & Applications

The common logarithms

The natural logarithms, the negative logarithms, basic laws of logarithms, real-life application of logarithms, practice questions, introduction to logarithms – explanation & examples.

The exponent of a number is the frequency or number of times a number is multiplied by itself. An expression that represents repeated multiplication of the same factor is called a power.

For example, the number 16 can be expressed in exponential form as; 2 4 . In this case, the numbers 2 and 4 are the base and exponent, respectively.

What is a Logarithm?

The concept of logarithm was introduced in the 17 th century by a Scottish mathematician named John Napier .

It was introduced to mechanical machinery in the 19 th century and to computers in the 20 th century. The natural logarithm is one of the useful functions in mathematics and has many applications.

Consider three numbers a, x and n, which are related as follows;

a x = M; where a > 0 < M and a ≠ 1

The number x is the logarithm of the number n to the base ‘a’. Therefore, a x = n can be expressed in logarithmic form as.

log a M = x, Here, M is the argument or the number; x is the exponent while ‘a’ is the base.

For example:

16 = 2 4 ⟹ log 2 16 = 4

9 = 3 2 ⟹ log 3 9 = 2 625 = 5 4 ⟹ log 5 625 = 4 7 0 = 1 ⟹ log 7 1 = 0 3 – 4 = 1/3 4 = 1/81 ⟹ log 3 1/81 = -4

All the logarithms with base 10 are called common logarithms . Mathematically, the common log of a number x is written as:

log 10 x = log x

A natural logarithm is a special form of logarithms in which the base is mathematical constant e, where e is an irrational number and equal to 2.7182818…. Mathematically, the natural log of a number x is written as:

log e x = ln x

where the natural log or ln is the inverse of e .

The natural exponential function is given as:

Introduction to Logarithm Common and Natural

We know that logarithms are not defined for negative values.

Then what do we mean by the negative logarithms?

It means that the logarithm of the set of such numbers gives a negative result. All the numbers that lie between 0 and 1 have negative logarithms.

There are four basic rules of logarithms. These are:

Product rule.

The product of two logarithms with a common base is equal to the sum of individual logarithms.

⟹ log b (m n) = log b m + log b n.

Division rule

The division rule of logarithms states that the quotient of two logarithmic values with the same bases is equal to each logarithm’s difference.

⟹ log b (m/n) = log b m – log b n

The exponential rule of logarithms

This rule states that the logarithm of a number with a rational exponent is equal to the product of the exponent and its logarithm.

⟹ log b (m n ) = n log b m

Change of Base

⟹ log b a = log x a ⋅ log b x

⟹ log b a = log x a / log x b

NOTE: The logarithm of a number is always stated together with its base. If the base is not given, it is assumed to be 10.

Logarithms very useful in the field of science, technology, and mathematics.

Here are a few examples of real-life applications of logarithms.

Electronic calculators have logarithms to make our calculations much easier.
Logarithms are used in surveys and celestial navigation.
Logarithms can be used to calculate the level of noise in decibels.
Ratio active decay, acidity [PH] of a substance and Richter scale are all measured in logarithmic form.

Let’s solve a few problems involving logarithms.

Solve for x in log 2 (64) = x

Here, 2 is the base, x is the exponent and 64 is the number.

Let 2 x = 64

Express 64 to the base of 2.

2 x = 2 × 2 × 2 × 2 × 2 × 2 = 2 6

x = 6, therefore, log 2 64 = 6.

Find x in log 10 100 = x

100 = number

x = exponent

Therefore, 10 x = 100

Hence x = 2

But 100 = 10 * 10 = 10 2

Solve for k given, log 3 x = log 3 4 + log 3 7

By applying the product rule log b (m n) = log b m + log b n we get;

⟹ log 3 4 + log 3 7= log 3 (4 * 7) = log 3 (28).

Hence, x = 28.

Solve for y given, log 2 x = 5

Here, 2 = base

5 = exponent

⟹ 2* 2 * 2 * 2 * 2 = 32

Thus, x = 32

Solve for log 10 105 given that, log 10 2 = 0.30103, log 10 3 = 0.47712 and log 10 7 = 0.84510

log 10 105 = log 10 (7 x 5 x 3)

Apply the product rule of logarithms = log 10 7 + log 10 5 + log 10 3 = log 10 7 + log 10 10/2 + log 10 3 = log 10 7 + log 10 10 – log 10 2 + log 10 3 = 0.845l0 + 1 – 0.30103 + 0.47712 = 2.02119.

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It may come as a surprise to many that often times mathematical concepts don't end up like they started! For those of you who think mathematics is timeless, fixed, and full of unchanging truths, such a proposition may seem unbelievable. But there are many instances in the history of mathematics of the development of a mathematical concept way beyond the purposes and potentialities that its original inventors intended. An example that will be familiar to you all is the logarithm.

What is a logarithm? Ask a modern mathematician nowadays and you will get a very different answer from the one you might have got from a mathematician several centuries ago. Indeed, even the very first mathematicians who worked with the logarithmic relation would have given an explanation that would seem quite foreign to a modern mathematician. So how did the logarithmic relation come about, and how is it that the concept underwent so much change? We will address these questions by looking at the emergence of this concept, and examining some of the issues surrounding its origins.

In fact, the question of the origins of the logarithmic relation does not have a simple answer. At least two scholars, the Scottish baron John Napier (1550-1617) and Swiss craftsman Joost Bürgi (1552-1632), produced independently systems that embodied the logarithmic relation and, within years of one another, produced tables for its use. This parallel insight is fascinating and rich in historical detail, and it reveals some methodological challenges for historians of mathematics. In light of all this, we will examine the ideas of these two scholars, as well as explore how historians have portrayed this intricate situation and the questions it raises about mathematics.

In large part, we intend to re-introduce teachers to a concept that is often taught without any reference to its original appearance on the mathematical scene. We hope that a close examination of Napier's and Bürgi's conceptions will enable teachers to consider alternative placement for introducing the idea of logarithms – as part of or after a unit on sequences. Furthermore, we provide in what follows mathematical and historical content, as well as student exercises, to promote the teaching of the logarithmic relation from its historical roots, which are firmly situated in simultaneous consideration of arithmetic and geometric sequences.

Kathleen M. Clark (The Florida State University) and Clemency Montelle (University of Canterbury), "Logarithms: The Early History of a Familiar Function - Introduction," Convergence (January 2011), DOI:10.4169/loci003495

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Logarithms: The Early History of a Familiar Function

Logarithms: The Early History of a Familiar Function - Logarithms: A 'Great Tale' for Use in the Classroom
Logarithms: The Early History of a Familiar Function - Before Logarithms: The Computational Demands of the Late Sixteenth Century
Logarithms: The Early History of a Familiar Function - John Napier Introduces Logarithms
Logarithms: The Early History of a Familiar Function - Joost Bürgi Introduces Logarithms
Logarithms: The Early History of a Familiar Function - The Challenges of Parallel Insights in the History of Mathematics
Logarithms: The Early History of a Familiar Function - Parallel Insights and the Reception of Mathematical Ideas
Logarithms: The Early History of a Familiar Function - Conclusion
Logarithms: The Early History of a Familiar Function - Appendix: Student Tasks
Logarithms: The Early History of a Familiar Function - Bibliography
Logarithms: The Early History of a Familiar Function - About the Authors and More Information on Sources

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Real Life Scenario of Logarithm

Hi Students, welcome to Amans Maths Blogs ( AMB ) . In this article, you will get an idea of “Real Life Scenario of Logarithm”. Real life scenario of logarithms is one of the most crucial concepts in our life.

As we know, in our maths book of 9th-10th class, there is a chapter named LOGARITHM is a very interesting chapter and its questions are some types that are required techniques to solve.

Therefore, you must read this article “Real Life Application of Logarithms” carefully.

Table of Contents

Usage of Logarithms

Definition of logarithms:.

Logarithms, often abbreviated as “logs,” are mathematical functions that represent the inverse operation of exponentiation. In simpler terms, logarithms provide a means to reverse the process of raising a number to a certain power.

The logarithm of a number with respect to a given base is the exponent to which the base must be raised to produce that number. Mathematically, it can be represented as:

y = log a x, where y is the logarithm of x to the base a.

Logarithms are commonly used to solve exponential equations, find unknown exponents, and express large or small numbers in more manageable forms.

Understanding this fundamental definition sets the stage for exploring the myriad applications of logarithms across various fields, where their ability to simplify complex calculations and represent exponential relationships proves invaluable.

Apart from Real Life Application of Logarithms, you can also clear you following doubts:

Doubt 1 : Why the function f(x) = 0 x is NOT an exponential function.

Doubt 2 : Why the function g(x) = 1 x is NOT an exponential function.

Doubt 3 : Why the function h(x) = (–2) x is NOT an exponential function.

Doubt 4 : Why the base of logarithm cannot be negative, means a > 0?

Doubt 5 : Why the base of logarithm cannot be unity, means a ≠ 1?

Doubt 6 : Why the logarithm of any negative number is not defined, mean log(–2) is not defined?

Types of Logarithms:

Logarithms come in several types, each with its own base, commonly denoted as ‘ a’ . The most widely used types of logarithms include:

Common Logarithms (Base = 10)

Common logarithms have a base of 10 and are denoted as log ( x ) or log 10 ( x ) .

It means, if the base is not mentioned in log, then you need to use base as ’10’. Logarithms are extensively utilized in various scientific calculations and engineering applications.

For instance, common logarithms are employed in seismic studies to measure earthquake magnitudes on the Richter scale, where each increase of one unit represents a tenfold increase in seismic amplitude.

Natural Logarithms (Base = e)

Natural logarithms have a base of the mathematical constant ‘ e’ , approximately equal to 2.71828, and are denoted as ln ( x ) .

They arise naturally in exponential growth and decay problems, as well as in calculus and mathematical modeling.

Natural logarithms find applications in finance, biology, and physics, aiding in the analysis of exponential processes and growth rates.

Binary Logarithms (Base = 2)

Binary logarithms, also known as logarithms to the base 2, are denoted as log 2 ( x ) .

They are prevalent in computer science and information theory, particularly in analyzing binary data and calculating the complexity of algorithms.

Binary logarithms help quantify the number of bits required to represent integers or the efficiency of binary search algorithms.

Logarithmic Rules:

Understanding the fundamental rules governing logarithms is essential for effectively manipulating and simplifying expressions involving logarithmic functions. Here are some key logarithmic rules:

Logarithm Product Rules

The product rule states that the logarithm of a product is equal to the sum of the logarithms of the individual factors. Mathematically, it can be expressed as:

log a (mn ) = log a ( m ) + log a ( n )

This rule allows us to break down complex products into simpler logarithmic expressions, facilitating calculations and problem-solving.

Logarithm Quotient Rules

The quotient rule dictates that the logarithm of a quotient is equivalent to the difference of the logarithms of the numerator and denominator. In equation form:

log a (m / n ) = log a ( m ) – log a ( n )

This rule enables us to decompose division operations into more manageable logarithmic terms, simplifying computations.

Logarithm Power Rules

The power rule states that the logarithm of a number raised to a power is equal to the exponent multiplied by the logarithm of the base. Mathematically, it can be represented as:

log a (m n ) =n log a ( m )

This rule facilitates the handling of exponential expressions within logarithmic functions, allowing for efficient manipulation and evaluation.

Logarithm Change of Base Formula

The change of base formula provides a method for converting logarithms from one base to another. It states that the logarithm of a number in a certain base can be expressed as the quotient of logarithms in different bases. The formula is given by:

log a (x ) = log c ( x ) / log c ( a )

where ‘c’ can be any base. This formula is particularly useful when dealing with logarithms in bases other than the commonly encountered bases of 10 and e.

Logarithm to Exponential and Exponential to Logarithm:

Expressing logarithms to exponentials.

Logarithms can be expressed as exponentials using the definition of logarithms.

If y = log a ( x ) ,

then the logarithmic equation can be rewritten in exponential form as

Expressing Exponentials to Logarithms

Exponentials can be expressed as logarithms using the inverse relationship between logarithms and exponentials.

If x = a y ,

y = log a ( x ) .

Real Life Applications of Logarithms

From measuring the intensity of earthquakes to calculating the pH levels of substances, logarithms offer a powerful tool for simplifying complex calculations and understanding exponential relationships.

In this discussion, we will explore several compelling real-life applications of logarithms, demonstrating how this mathematical concept plays a crucial role in various fields such as science, finance, and technology.

Applications of Logarithms: Measuring Sound Intensity Using Decibel

Definition of decibels.

Decibels are a unit of measurement used to express the relative intensity of sound. The decibel scale is logarithmic, with each increase of 10 dB representing a tenfold increase in sound intensity.

Mathematically, the relationship between sound intensity ‘I’ and sound level in decibels L can be expressed as:

L = 10 . log 10 (I / I 0 )

where I 0 is the reference intensity level (usually set at the threshold of human hearing, approximately 1 0 −12 watts per square meter).

Applications in Noise Measurement:

Decibels are commonly used to measure noise levels in various environments, such as workplaces, urban areas, and transportation hubs.

By quantifying sound intensity on a logarithmic scale, decibels allow for a more nuanced assessment of noise pollution and its potential impact on human health and well-being.

Regulatory agencies often set noise exposure limits based on decibel measurements to mitigate the adverse effects of excessive noise.

Sound Engineering and Music Production:

In sound engineering and music production, decibels are used to calibrate audio equipment, determine optimal sound levels, and control the balance between different audio components.

Engineers and producers rely on decibel measurements to achieve desired sound quality, dynamic range, and clarity in recordings and live performances.

Decibels also play a crucial role in audio mastering and mixing processes, ensuring consistent sound levels across different tracks and audio sources.

Applications of Logarithms: Stock Market Analysis

Percentage changes and returns:.

Logarithms are commonly used to calculate percentage changes and returns in stock prices.

By taking the logarithm of the ratio of current price to previous price, analysts can determine the logarithmic return, which represents the percentage change in price over a given period.

Logarithmic returns provide a more symmetrical and interpretable measure of price changes, particularly for volatile assets, and are widely used in financial modeling and risk management.

Volatility Measurement:

Logarithms are instrumental in measuring volatility, a key parameter in assessing risk and investment performance. Volatility, often quantified using standard deviation or variance, reflects the magnitude of price fluctuations in financial markets.

Logarithmic returns are used to calculate volatility metrics such as historical volatility and implied volatility, which help investors gauge the level of uncertainty and potential price movements in the market.

Technical Analysis:

In technical analysis, logarithms are employed to transform price series and indicators to achieve stationarity and normalize data distributions.

Common transformations include taking the natural logarithm of prices or volume measures to stabilize variance and remove trends.

Logarithmic transformations facilitate the application of statistical techniques and pattern recognition algorithms, enabling traders to identify trends, support and resistance levels, and trading signals more effectively.

Portfolio Optimization:

Logarithms are utilized in portfolio optimization techniques to model asset returns and correlations.

By transforming asset returns into logarithmic space, analysts can linearize the relationship between asset returns and portfolio returns, making optimization models more tractable.

Logarithmic transformations also allow for the application of mean-variance analysis and modern portfolio theory, guiding investors in constructing diversified portfolios that balance risk and return objectives.

Applications of Logarithms in Decay of Radioactive Elements

Logarithms are fundamental in understanding and modeling the decay of radioactive elements, a process governed by exponential decay kinetics. Here’s how logarithms are applied in the study of radioactive decay:

The decay of radioactive elements follows a predictable pattern described by the decay law. The decay rate, commonly denoted as λ , represents the probability of a radioactive nucleus decaying per unit time.

The number of radioactive nuclei remaining after a certain time t can be expressed using the exponential decay equation:

N(t) = N 0 . e – λt ,

where N ( t ) is the number of radioactive nuclei at time t, N 0 is the initial number of nuclei, and e is the base of the natural logarithm. Logarithms come into play when solving for unknown variables such as decay rate or time elapsed, as well as interpreting experimental data.

Half Life Calculuation:

Logarithms are instrumental in determining the half-life ( T 1/2 ) of a radioactive element, which is the time required for half of the radioactive nuclei to decay. The half-life can be derived from the decay constant ( λ ) using the relationship:

T1/2 = ln(2) / λ

By taking the natural logarithm of both sides of the decay equation, scientists can isolate the decay constant and calculate the half-life, a crucial parameter in radiometric dating and nuclear medicine.

Activity Measurement:

Logarithms are used to quantify the activity of radioactive samples, which represents the rate of decay or the number of decays per unit time.

The activity ( A ) of a radioactive sample can be calculated using the equation: A = λ . N

where λ is the decay constant and N is the number of radioactive nuclei.

Logarithmic transformations enable scientists to convert activity measurements into meaningful units such as becquerels (Bq) or curies (Ci), facilitating comparisons and regulatory compliance.

Applications of Logarithms in Radiocarbon Dating

Radiocarbon dating is a widely used technique for determining the age of archaeological artifacts and geological samples containing organic material.

Logarithms play a crucial role in radiocarbon dating calculations and interpretations. Here’s how logarithms are applied in radiocarbon dating:

Decay of Carbon-14:

Radiocarbon dating relies on the radioactive decay of carbon-14 ( 14 C ), a naturally occurring isotope of carbon.

Carbon-14 undergoes radioactive decay by emitting a beta particle, transforming into nitrogen-14 ( 14 N ).

The decay process follows exponential decay kinetics, where the rate of decay is proportional to the number of radioactive nuclei present.

Half-Life of Carbon-14:

The half-life of carbon-14 is approximately 5,730 years, meaning that it takes 5,730 years for half of the carbon-14 atoms in a sample to decay into nitrogen-14.

Logarithms are employed to calculate the age of a sample based on the ratio of carbon-14 to stable carbon-12 ( 12 C ) isotopes.

By measuring the remaining ratio of carbon-14 to carbon-12 and applying the exponential decay equation, scientists can determine the age of the sample.

Libby’s Law:

Radiocarbon dating calculations are based on Libby’s Law, which relates the activity of carbon-14 in a sample to its age.

The activity ( A ) of carbon-14 is proportional to the number of radioactive nuclei present, following the equation:

A(t) = A 0 . e – λt ,

where A 0 is the initial activity of the sample, t is the age of the sample, and λ is the decay constant of carbon-14.

Logarithms are used to solve for the age of the sample (t) by rearranging the equation and isolating the variable of interest.

Application of Logarithms in Measuring Earthquakes Using the Richter Scale:

The Richter scale, developed by Charles F. Richter in 1935, is a logarithmic scale used to measure the magnitude of earthquakes.

Logarithms play a crucial role in quantifying and comparing the seismic energy released by earthquakes. Here’s how logarithms are applied in measuring earthquakes using the Richter scale:

Definition of Magnitude:

The magnitude of an earthquake is a measure of the energy released at the earthquake’s source.

The Richter scale measures this energy logarithmically, with each whole number increase on the Richter scale representing a tenfold increase in amplitude of seismic waves and approximately 31.6 times more energy release.

Mathematically, the Richter magnitude M of an earthquake is calculated using the equation: M = 10 . log 10 (A / A 0 ),

where A is the maximum amplitude of seismic waves recorded on a seismogram and A 0 is a reference amplitude.

For example: if we note the magnitude of the earthquake on the Richter scale as 2, then the other next magnitude on the scale is explained in the following table.

Now according to the Richter scale magnitude of the earthquake, there is a lot of bad effect on our environments which may be a danger to the real world. Its details are given below in the table.

This is one of the real-life scenario of logarithms, which must be known.

Logarithmic Representation:

Logarithms allow for the concise and intuitive representation of seismic energy on a logarithmic scale.

By taking the logarithm of the ratio of seismic wave amplitudes, the Richter magnitude compresses a wide range of energy levels into a manageable scale.

This logarithmic representation enables scientists to compare earthquakes of different magnitudes and assess their relative impact on structures and communities.

Sensitivity to Energy Release:

The logarithmic nature of the Richter scale makes it highly sensitive to small changes in seismic energy. Even minor increases in earthquake magnitude result in significant jumps in the measured Richter magnitude.

For example, a magnitude 6.0 earthquake releases ten times more energy than a magnitude 5.0 earthquake, while a magnitude 7.0 earthquake releases a thousand times more energy than a magnitude 5.0 earthquake. Logarithmic scaling ensures that the Richter scale captures the full range of seismic energy released by earthquakes.

Application of Logarithms in Measuring pH Levels of Chemicals

The pH scale is a logarithmic scale used to measure the acidity or alkalinity of a solution.

Logarithms play a fundamental role in quantifying pH levels and understanding the behavior of acidic and basic substances.

Here’s how logarithms are applied in measuring pH levels of chemicals:

Definition of pH:

pH is a measure of the concentration of hydrogen ions ( H + ) in a solution.

The pH scale ranges from 0 to 14, with a pH of 7 considered neutral, pH values below 7 indicating acidity, and pH values above 7 indicating alkalinity.

The pH of a solution is calculated using the negative logarithm of the hydrogen ion concentration ( [ H + ] ) in moles per liter ( mol/L mol/L ):

pH = -log 10 [H + ]

Logarithms are essential for transforming the hydrogen ion concentration into a more manageable scale, enabling the quantification of acidity and alkalinity on a numerical scale.

Logarithmic pH Scaling:

Logarithms compress the wide range of hydrogen ion concentrations found in solutions into a scale that is easily interpreted and compared.

Each unit change in pH represents a tenfold change in hydrogen ion concentration.

For example, a solution with a pH of 5 has 10 times more hydrogen ions than a solution with a pH of 6 and 100 times more hydrogen ions than a solution with a pH of 7.

Logarithmic scaling ensures that small changes in acidity or alkalinity result in significant changes in pH, making the scale sensitive to variations in solution chemistry.

Practical Applications:

The logarithmic nature of the pH scale has numerous practical applications in various industries and fields. In chemistry, pH measurements are crucial for assessing the reactivity and stability of chemical compounds, controlling chemical processes, and maintaining optimal conditions for biological systems.

In environmental science, pH measurements are used to monitor water quality, assess pollution levels, and study the health of aquatic ecosystems. In agriculture, pH measurements guide soil management practices and fertilizer application, ensuring optimal nutrient availability for plant growth.

Application of Logarithms in Calculating Population Growth:

Logarithms are instrumental in modeling and analyzing the growth of human populations and other living species. Through population dynamics models, logarithms provide valuable insights into population growth rates, carrying capacities, and sustainability.

Here’s how logarithms are applied in calculating the growth of human species or other living species:

Exponential Growth Model:

Logarithms are closely associated with exponential growth, a common model used to describe population growth in ideal conditions.

The exponential growth model assumes that populations grow at a constant percentage rate over time. Mathematically, the exponential growth equation is expressed as:

N(t) = N 0 . e r t ,

where N ( t ) is the population size at time t, N 0 is the initial population size, r is the per capita growth rate, and e is the base of the natural logarithm.

Logarithms enable scientists to transform exponential growth equations into linear forms for analysis and interpretation.

Logistic Growth Model:

The logistic growth model incorporates limitations on population growth, such as finite resources and environmental constraints.

Unlike exponential growth, logistic growth accounts for carrying capacity, the maximum population size that a habitat can support sustainably. The logistic growth equation is given by

N(t) = K / [1 + (K – N 0 ) / N 0 * e -rt ]

where K is the carrying capacity of the environment.

Logarithms play a role in solving logistic growth equations and analyzing population dynamics near carrying capacity.

Doubling Time and Half-Life:

Logarithms are used to calculate doubling time and half-life in population growth scenarios.

Doubling time represents the time required for a population to double in size under exponential growth conditions.

It is calculated using the formula:

T double = ln(2) / r

where r is the growth rate. Similarly, half-life represents the time required for a population to reach half of its maximum size under logistic growth conditions. It is calculated using logarithms and the logistic growth equation.

Applications of Logarithms in Analyzing Drug Concentration:

Logarithms play a crucial role in pharmacokinetics, the study of how drugs are absorbed, distributed, metabolized, and excreted in the body.

Logarithmic transformations are commonly used to analyze drug concentration data and understand the pharmacological behavior of medications.

Here’s how logarithms are applied in analyzing drug concentration in medicines:

Pharmacokinetic Models:

Pharmacokinetic models describe the time course of drug concentration in the body following administration.

These models typically involve exponential decay (for drug elimination) or exponential increase (for drug absorption) of drug concentration over time.

Logarithmic transformations are used to linearize pharmacokinetic data and facilitate the estimation of pharmacokinetic parameters such as clearance, volume of distribution, and half-life.

Linear Pharmacokinetic Models:

In linear pharmacokinetics, drug concentration changes proportionally with dose and time.

Logarithmic transformations are applied to linearize drug concentration-time profiles, allowing for the calculation of pharmacokinetic parameters using linear regression analysis.

For example, the elimination rate constant ( k el ) can be estimated from the slope of the linear portion of the drug concentration-time curve, which is obtained after logarithmic transformation.

Non-Linear Pharmacokinetic

In nonlinear pharmacokinetics, drug concentration changes non-proportionally with dose or time due to saturation of drug-metabolizing enzymes or transporters.

Logarithmic transformations are used to transform nonlinear pharmacokinetic data into linear form, enabling the application of linear regression techniques to estimate parameters such as Michaelis-Menten constants ( K m ) and maximum rate of metabolism ( V max ).

Drug Bioavailability and Bioequivalence:

Logarithmic transformations are employed in the assessment of drug bioavailability and bioequivalence.

Bioavailability measures the fraction of an administered drug that reaches systemic circulation unchanged, while bioequivalence compares the pharmacokinetic parameters of different formulations of the same drug.

Logarithmic transformations are used to assess the extent and rate of drug absorption, enabling comparisons between different formulations or routes of administration.

Applications of Logarithms in Calculating Complex Values

Sometimes we need to find the values of some complex calculations like x = (31)^(1/5) (5th root of 31), finding a number of digits in the values of (12)^256 etc. To solve these types of problems, we need to use the logarithms.

The solving method of these problems will be learning in another maths blogs post. The URL of the post will be mentioned below in the future.

Practice: Logarithm Questions Set 1

Applications of Logarithms in Compound Interest:

Logarithms are fundamental in understanding and calculating compound interest, a concept central to finance and investment.

Compound interest refers to the interest earned not only on the initial principal amount but also on the accumulated interest from previous periods.

Logarithms play a crucial role in determining the future value of investments, amortizing loans, and evaluating investment performance.

Here’s how logarithms are applied in compound interest calculations:

Compound Interest Formula:

The compound interest formula calculates the future value ( F V ) of an investment or loan, taking into account the initial principal ( P ), the annual interest rate ( r ), the number of compounding periods per year ( n ), and the time in years ( t ). The formula is given by:

FV = P x (1 + r/100) nt

Logarithms are used to solve for any unknown variable in the compound interest formula, such as the principal, interest rate, compounding frequency, or time.

Continuous Compounding:

Logarithms facilitate the calculation of compound interest under continuous compounding, where interest is compounded infinitely often over time. The formula for continuous compounding is:

FV = P x e rt

where e is the base of the natural logarithm. Logarithms enable the transformation of continuous compounding equations into more manageable forms and allow for comparisons with compound interest calculations using discrete compounding periods.

Time Value of Money:

Logarithms are used to calculate the present value ( P V ) of future cash flows or the future value ( F V ) of present investments, taking into account the time value of money.

By discounting future cash flows back to their present value using logarithmic transformations, investors can make informed decisions about the profitability and riskiness of investment opportunities.

Amortization of Loans:

Logarithms are employed in loan amortization schedules to calculate the periodic payments required to repay a loan over time.

By decomposing the total payment into principal and interest components using logarithmic transformations, borrowers can understand the distribution of payments and the impact of different repayment schedules on the total cost of borrowing.

Difference Between Algorithms and Logarithms

Algorithms and logarithms are both fundamental concepts in mathematics and computer science, but they serve different purposes and have distinct characteristics. Here’s a comparison between algorithms and logarithms:

Algorithms vs Logarithms: Difference 1#

Algorithm : An algorithm is a step-by-step procedure or set of rules for solving a problem or performing a specific task. Algorithms can be expressed in natural language, pseudocode, or programming languages and are used to automate processes and achieve desired outcomes.

Logarithm : A logarithm is a mathematical function that represents the exponent to which a fixed number, called the base, must be raised to produce a given number. Logarithms are used to solve exponential equations, transform data, and quantify relationships between variables.

Algorithms vs Logarithms: Difference 2#

Algorithm : Algorithms are used to solve computational problems, automate tasks, and optimize processes. They are widely used in computer science, engineering, data analysis, and various other fields to perform calculations, manipulate data, and make decisions.

Logarithm : Logarithms are used primarily in mathematics and science to simplify calculations, express relationships between variables, and solve exponential equations. They have applications in fields such as finance, chemistry, physics, and biology for modeling natural phenomena and analyzing data.

Algorithms vs Logarithms: Difference 3#

Algorithm : Algorithms can be represented in various forms, including pseudocode, flowcharts, and programming code. They specify a sequence of steps or instructions to be followed to solve a problem or achieve a specific objective.

Logarithm : Logarithms are represented using mathematical notation, typically written as ( log b ( x )) , where b is the base and x is the number whose logarithm is being calculated. Common bases include 10 (common logarithm) and e (natural logarithm).

Algorithms vs Logarithms: Difference 4#

Algorithm : Algorithms are used to design software applications, develop algorithms for solving complex problems, implement data structures and algorithms, and optimize processes in various domains.

Logarithm : Logarithms are used in mathematical calculations, scientific research, engineering analysis, financial modeling, and statistical analysis. They help simplify mathematical expressions, quantify relationships between variables, and solve exponential equations.

FAQ (Frequently Asked Questions) on Real Life Application of Logarithms

How are logarithms used in engineering.

In engineering, all the two types of logarithms known as common logarithm and natural logarithm, are used. In chemical engineering, logarithms are used to measure radioactive decay and pH solutions. In biomedical engineering, logarithms are used to measure cell decay and growth.

How are logarithms used in finance?

In finance, the logarithms is used in quantitative finance (specially in CFA Level 1, 2, 3 Exams ). Here, the base of the logarithm is e (exponential). By using log of any number on base e, we can compute continuously compounded returns.

What is log10 equal to?

If the base of the log is 10, then the value of log 10 10 = 1 using the formula of log a a = 1. If the base of the log is e, then the value of log e 10 = 2.302585.

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What are logarithms and why are they useful? Get the basics on these critical mathematical functions -- and discover why smart use of logarithms can determine whether your eyes turn red at the swimming pool this summer. Lesson by Steve Kelly, animation by TED-Ed.

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Essay on how logarithms used in real life?

In mathematics, a logarithm is the exponent to which a number must be raised in order to obtain a given number. For example, the base-ten logarithm of 100 is 2, because ten raised to the power of two equals 100: log10 100 = 2 But what does this have to do with real life? It turns out that logarithms are used in a variety of ways, from predicting population growth to measuring the decibels of sounds!

What are logarithms?

A logarithm is a mathematical function that allows us to express a number in terms of another number. For example, the logarithm of 100 is 2, because 100 can be expressed as 10 to the power of 2. Logarithms are useful in many fields, including mathematics, engineering, and physics.

In mathematics, logarithms are used to solve equations. In engineering, they are used to design circuits. In physics, they are used to calculate the strength of waves.

Logarithms are also used in real life . For example, they can be used to measure the height of a building. To do this, you would need to know the angle of elevation of the top of the building and the distance from the base of the building to the point where you are measuring.

What are some real life applications of logarithms?

One of the most common applications of logarithms is in calculating exponential growth. For example, when determining how long it will take for an investment to double, the Rule of 72 can be applied. This rule states that the time it takes for an investment to double is approximately equal to 72 divided by the interest rate. Thus, if you have an investment that is earning 8% interest, it would take approximately 9 years (72/8) for it to double.

Logarithms can also be used to calculate compound interest. Compound interest occurs when interest is earned not only on the original investment, but also on the accumulated interest from previous periods. This can be written as an exponential equation, with the amount of money after t years given by:

A = P(1 + r/n)^nt

where A is the final amount, P is the original principal (the amount of money invested), r is the annual interest rate, n is the number of compounding periods per year, and t is the number of years. Thus, we can see that compound interest is simply a case of exponential growth.

Finally, logarithms can also be used in decibel calculations. The decibel (dB) is a unit used to measure the loudness of a sound. The formula for calculating the dB level of a sound is:

dB = 10 log10(I/I0)

where I is the intensity of the sound and I0 is the reference intensity. The reference intensity is typically taken to be 10^-12 Watts/m^2, which is the threshold of human hearing. Thus, we can see that the dB level of a sound is simply the logarithm of the ratio of the sound’s intensity to the reference intensity.

How to calculate logarithms?

Most people don’t realize that logarithms are used in everyday life. Here are a few examples of how you can use them to make calculations:

1. To calculate the number of digits in a number, take the logarithm of the number and add 1. For example, the number 125 has 3 digits, so the logarithm of 125 (log 125) + 1 = 3.

2. To find out what exponent to raise a number to in order to get a specific result, take the logarithm of the result and divide it by the logarithm of the number. For example, if you want to know what exponent you need to raise 2 to in order to get 8, take the logarithm of 8 (log 8) and divide it by the logarithm of 2 (log 2). The answer is 3, so 2 raised to the 3rd power is 8.

3. To calculate compound interest, use this formula: A = P(1 + r/n)^nt. In this formula, A is the amount of money after n years, P is the principal (the original amount), r is the interest rate (expressed as a decimal), and n is the number of compounding periods per year. If you take the logarithm of both sides of this equation, you get: log A = log P + ntlog(1 + r/n). This can be simplified to: log A = log P + ntlog(1 + r) – ntlog(1 + r/n).

Logarithms are mathematical tools that are used in a variety of real-world applications. From calculating the properties of musical instruments to measuring the intensity of earthquakes, logarithms play a vital role in many different fields. As we have seen, they can be used to solve complex problems and help us to understand the world around us better.

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6.5: Applications of Exponential and Logarithmic Functions

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Carl Stitz & Jeff Zeager
Lakeland Community College & Lorain County Community College

As we mentioned in Section 6.1 , exponential and logarithmic functions are used to model a wide variety of behaviors in the real world. In the examples that follow, note that while the applications are drawn from many different disciplines, the mathematics remains essentially the same. Due to the applied nature of the problems we will examine in this section, the calculator is often used to express our answers as decimal approximations.

6.5.1. Applications of Exponential Functions

Perhaps the most well-known application of exponential functions comes from the financial world. Suppose you have $\$ 100$ to invest at your local bank and they are offering a whopping $5 \, \%$ annual percentage interest rate. This means that after one year, the bank will pay you $5 \%$ of that $\$100$, or $\$ 100(0.05) =\$ 5$ in interest, so you now have $\$105$. 1 This is in accordance with the formula for simple interest which you have undoubtedly run across at some point before.

Equation 6.1. Simple Interest

The amount of interest $I$ accrued at an annual rate $r$ on an investment a $P$ after $t$ years is \[I = Prt\nonumber\] The amount $A$ in the account after $t$ years is given by \[A = P + I = P + Prt = P(1+rt)\nonumber\]

Suppose, however, that six months into the year, you hear of a better deal at a rival bank. 2 Naturally, you withdraw your money and try to invest it at the higher rate there. Since six months is one half of a year, that initial $\$100$ yields $\$100(0.05)\left(\frac{1}{2}\right) = \$ 2.50$ in interest. You take your $\$102.50$ off to the competitor and find out that those restrictions which may apply actually apply to you, and you return to your bank which happily accepts your $\$102.50$ for the remaining six months of the year. To your surprise and delight, at the end of the year your statement reads $\$105.06$, not $\$105$ as you had expected. 3 Where did those extra six cents come from? For the first six months of the year, interest was earned on the original principal of $\$100$, but for the second six months, interest was earned on $\$102.50$, that is, you earned interest on your interest. This is the basic concept behind compound interest . In the previous discussion, we would say that the interest was compounded twice, or semiannually. 4 If more money can be earned by earning interest on interest already earned, a natural question to ask is what happens if the interest is compounded more often, say $4$ times a year, which is every three months, or ‘quarterly.’ In this case, the money is in the account for three months, or $\frac{1}{4}$ of a year, at a time. After the first quarter, we have $A = P(1+rt) = \$100 \left(1 + 0.05 \cdot \frac{1}{4} \right) = \$101.25$. We now invest the $\$101.25$ for the next three months and find that at the end of the second quarter, we have $A = \$101.25 \left(1 + 0.05 \cdot \frac{1}{4} \right)\approx \$102.51$. Continuing in this manner, the balance at the end of the third quarter is $\$103.79$, and, at last, we obtain $\$105.08$. The extra two cents hardly seems worth it, but we see that we do in fact get more money the more often we compound. In order to develop a formula for this phenomenon, we need to do some abstract calculations. Suppose we wish to invest our principal $P$ at an annual rate $r$ and compound the interest $n$ times per year. This means the money sits in the account $\frac{1}{n}^{\mbox{\tiny th}}$ of a year between compoundings. Let $A_{k}$ denote the amount in the account after the $k^{\mbox{\tiny th}}$ compounding. Then $A_{1} = P\left(1 + r\left(\frac{1}{n}\right)\right)$ which simplifies to $A_{1} = P \left(1 + \frac{r}{n}\right)$. After the second compounding, we use $A_{1}$ as our new principal and get $A_{2} = A_{1} \left(1 + \frac{r}{n}\right) = \left[P \left(1 + \frac{r}{n}\right)\right]\left(1 + \frac{r}{n}\right) = P \left(1 + \frac{r}{n}\right)^2$. Continuing in this fashion, we get $A_{3} =P \left(1 + \frac{r}{n}\right)^3$, $A_{4} =P \left(1 + \frac{r}{n}\right)^4$, and so on, so that $A_{k} = P \left(1 + \frac{r}{n}\right)^k$. Since we compound the interest $n$ times per year, after $t$ years, we have $nt$ compoundings. We have just derived the general formula for compound interest below.

Equation 6.2. Compounded Interest

If an initial principal $P$ is invested at an annual rate $r$ and the interest is compounded $n$ times per year, the amount $A$ in the account after $t$ years is \[A(t) = P \left(1 + \frac{r}{n}\right)^{nt}\nonumber\]

If we take $P = 100$, $r = 0.05$, and $n = 4$, Equation 6.2 becomes $A(t) = 100\left(1+ \frac{0.05}{4}\right)^{4t}$ which reduces to $A(t) = 100(1.0125)^{4t}$. To check this new formula against our previous calculations, we find $A\left(\frac{1}{4}\right) = 100(1.0125)^{4 \left(\frac{1}{4}\right)} = 101.25$, $A\left(\frac{1}{2}\right) \approx \$102.51$, $A\left(\frac{3}{4}\right) \approx \$103.79$, and $A(1) \approx \$105.08$.

Example 6.5.1

Suppose $\$2000$ is invested in an account which offers $7.125 \%$ compounded monthly.

Express the amount $A$ in the account as a function of the term of the investment $t$ in years.
How much is in the account after $5$ years?
How long will it take for the initial investment to double?
Find and interpret the average rate of change 5 of the amount in the account from the end of the fourth year to the end of the fifth year, and from the end of the thirty-fourth year to the end of the thirty-fifth year.
Substituting $P = 2000$, $r = 0.07125$, and $n = 12$ (since interest is compounded monthly ) into Equation 6.2 yields $A(t) = 2000\left(1 + \frac{0.07125}{12}\right)^{12t}=2000 (1.0059375)^{12t}$.
Since $t$ represents the length of the investment in years, we substitute $t=5$ into $A(t)$ to find $A(5) = 2000 (1.0059375)^{12(5)} \approx 2852.92$. After $5$ years, we have approximately $\$2852.92$.
Our initial investment is $\$2000$, so to find the time it takes this to double, we need to find $t$ when $A(t) = 4000$. We get $2000 (1.0059375)^{12t}=4000$, or $(1.0059375)^{12t}=2$. Taking natural logs as in Section 6.3 , we get $t = \frac{\ln(2)}{12 \ln(1.0059375)} \approx 9.75$. Hence, it takes approximately $9$ years $9$ months for the investment to double.
To find the average rate of change of $A$ from the end of the fourth year to the end of the fifth year, we compute $\frac{A(5)-A(4)}{5-4} \approx 195.63$. Similarly, the average rate of change of $A$ from the end of the thirty-fourth year to the end of the thirty-fifth year is $\frac{A(35)-A(34)}{35-34} \approx 1648.21$. This means that the value of the investment is increasing at a rate of approximately $\$195.63$ per year between the end of the fourth and fifth years, while that rate jumps to $\$1648.21$ per year between the end of the thirty-fourth and thirty-fifth years. So, not only is it true that the longer you wait, the more money you have, but also the longer you wait, the faster the money increases. 6

We have observed that the more times you compound the interest per year, the more money you will earn in a year. Let’s push this notion to the limit. 7 Consider an investment of $\$ 1$ invested at $100 \%$ interest for $1$ year compounded $n$ times a year. Equation 6.2 tells us that the amount of money in the account after $1$ year is $A = \left(1+\frac{1}{n}\right)^{n}$. Below is a table of values relating $n$ and $A$.

\[\begin{array}{|r||r|} \hline n & A \\ \hline 1 & 2 \\ \hline 2 & 2.25 \\ \hline 4 & \approx 2.4414 \\ \hline 12 & \approx 2.6130 \\ \hline 360 & \approx 2.7145 \\ \hline 1000 & \approx 2.7169 \\ \hline 10000 & \approx 2.7181 \\ \hline 100000 & \approx 2.7182 \\ \hline \end{array}\nonumber\]

As promised, the more compoundings per year, the more money there is in the account, but we also observe that the increase in money is greatly diminishing. We are witnessing a mathematical ‘tug of war’. While we are compounding more times per year, and hence getting interest on our interest more often, the amount of time between compoundings is getting smaller and smaller, so there is less time to build up additional interest. With Calculus, we can show 8 that as $n \rightarrow \infty$, $A = \left(1+\frac{1}{n}\right)^{n} \rightarrow e$, where $e$ is the natural base first presented in Section 6.1 . Taking the number of compoundings per year to infinity results in what is called continuously compounded interest.

Theorem 6.8.

If you invest $\$1$ at $100 \%$ interest compounded continuously, then you will have $\$ e$ at the end of one year.

Using this definition of $e$ and a little Calculus, we can take Equation 6.2 and produce a formula for continuously compounded interest.

Equation 6.3. Continuously Compounded Interest

If an initial principal $P$ is invested at an annual rate $r$ and the interest is compounded continuously, the amount $A$ in the account after $t$ years is \[A(t) = P e^{rt}\nonumber\]

If we take the scenario of Example 6.5.1 and compare monthly compounding to continuous compounding over $35$ years, we find that monthly compounding yields $A(35) = 2000 (1.0059375)^{12(35)}$ which is about $\$ 24,\!035.28$, whereas continuously compounding gives $A(35) = 2000e^{0.07125 (35)}$ which is about $\$ 24,\!213.18$ - a difference of less than $1 \%$.

Equations 6.2 and 6.3 both use exponential functions to describe the growth of an investment. Curiously enough, the same principles which govern compound interest are also used to model short term growth of populations. In Biology, The Law of Uninhibited Growth states as its premise that the instantaneous rate at which a population increases at any time is directly proportional to the population at that time. 9 In other words, the more organisms there are at a given moment, the faster they reproduce. Formulating the law as stated results in a differential equation, which requires Calculus to solve. Its solution is stated below.

Equation 6.4. Uninhibited Growth

If a population increases according to The Law of Uninhibited Growth, the number of organisms $N$ at time $t$ is given by the formula \[N(t) = N_0e^{kt},\nonumber\] where $N(0) = N_0$ (read ‘$N$ nought’) is the initial number of organisms and $k>0$ is the constant of proportionality which satisfies the equation

\[\left(\mbox{instantaneous rate of change of $N(t)$ at time $t$}\right) = k \, N(t)\nonumber\]

It is worth taking some time to compare Equations 6.3 and 6.4 . In Equation 6.3 , we use $P$ to denote the initial investment; in Equation 6.4 , we use $N_0$ to denote the initial population. In Equation 6.3 , $r$ denotes the annual interest rate, and so it shouldn’t be too surprising that the $k$ in Equation 6.4 corresponds to a growth rate as well. While Equations 6.3 and 6.4 look entirely different, they both represent the same mathematical concept.

Example 6.5.2

In order to perform arthrosclerosis research, epithelial cells are harvested from discarded umbilical tissue and grown in the laboratory. A technician observes that a culture of twelve thousand cells grows to five million cells in one week. Assuming that the cells follow The Law of Uninhibited Growth, find a formula for the number of cells, $N$, in thousands, after $t$ days.

We begin with $N(t) = N_0e^{kt}$. Since $N$ is to give the number of cells in thousands , we have $N_0 = 12$, so $N(t) = 12e^{kt}$. In order to complete the formula, we need to determine the growth rate $k$. We know that after one week, the number of cells has grown to five million. Since $t$ measures days and the units of $N$ are in thousands, this translates mathematically to $N(7) = 5000$. We get the equation $12e^{7k} = 5000$ which gives $k = \frac{1}{7} \ln\left(\frac{1250}{3}\right)$. Hence, $N(t) = 12e^{ \frac{t}{7} \ln\left(\frac{1250}{3}\right)}$. Of course, in practice, we would approximate $k$ to some desired accuracy, say $k \approx 0.8618$, which we can interpret as an $86.18 \%$ daily growth rate for the cells.

Whereas Equations 6.3 and 6.4 model the growth of quantities, we can use equations like them to describe the decline of quantities. One example we’ve seen already is Example 6.1.1 in Section 6.1 . There, the value of a car declined from its purchase price of $\$25,\!000$ to nothing at all. Another real world phenomenon which follows suit is radioactive decay. There are elements which are unstable and emit energy spontaneously. In doing so, the amount of the element itself diminishes. The assumption behind this model is that the rate of decay of an element at a particular time is directly proportional to the amount of the element present at that time. In other words, the more of the element there is, the faster the element decays. This is precisely the same kind of hypothesis which drives The Law of Uninhibited Growth, and as such, the equation governing radioactive decay is hauntingly similar to Equation 6.4 with the exception that the rate constant $k$ is negative.

Equation 6.5. Radioactive Decay

The amount of a radioactive element $A$ at time $t$ is given by the formula \[A(t) = A_0e^{kt},\nonumber\] where $A(0) = A_0$ is the initial amount of the element and $k<0$ is the constant of proportionality which satisfies the equation

\[\left(\mbox{instantaneous rate of change of $A(t)$ at time $t$}\right) = k \, A(t)\nonumber\]

Example 6.5.3

Iodine-131 is a commonly used radioactive isotope used to help detect how well the thyroid is functioning. Suppose the decay of Iodine-131 follows the model given in Equation 6.5 , and that the half-life 10 of Iodine-131 is approximately $8$ days. If $5$ grams of Iodine-131 is present initially, find a function which gives the amount of Iodine-131, $A$, in grams, $t$ days later.

Since we start with $5$ grams initially, Equation 6.5 gives $A(t) = 5e^{kt}$. Since the half-life is $8$ days, it takes $8$ days for half of the Iodine-131 to decay, leaving half of it behind. Hence, $A(8) = 2.5$ which means $5e^{8k} = 2.5$. Solving, we get $k = \frac{1}{8} \ln\left(\frac{1}{2}\right) = -\frac{\ln(2)}{8} \approx -0.08664$, which we can interpret as a loss of material at a rate of $8.664 \%$ daily. Hence, $A(t) = 5 e^{-\frac{t\ln(2)}{8}} \approx 5 e^{-0.08664t}$.

We now turn our attention to some more mathematically sophisticated models. One such model is Newton’s Law of Cooling, which we first encountered in Example 6.1.2 of Section 6.1 . In that example we had a cup of coffee cooling from $160^{\circ}\mbox{F}$ to room temperature $70^{\circ}\mbox{F}$ according to the formula $T(t) = 70 + 90 e^{-0.1 t}$, where $t$ was measured in minutes. In this situation, we know the physical limit of the temperature of the coffee is room temperature, 11 and the differential equation which gives rise to our formula for $T(t)$ takes this into account. Whereas the radioactive decay model had a rate of decay at time $t$ directly proportional to the amount of the element which remained at time $t$, Newton’s Law of Cooling states that the rate of cooling of the coffee at a given time $t$ is directly proportional to how much of a temperature exists between the coffee at time $t$ and room temperature, not the temperature of the coffee itself. In other words, the coffee cools faster when it is first served, and as its temperature nears room temperature, the coffee cools ever more slowly. Of course, if we take an item from the refrigerator and let it sit out in the kitchen, the object’s temperature will rise to room temperature, and since the physics behind warming and cooling is the same, we combine both cases in the equation below.

Equation 6.6. Newton’s Law of Cooling (Warming)

The temperature $T$ of an object at time $t$ is given by the formula \[T(t) = T_{a} + \left(T_0 - T_{a}\right) e^{-kt},\nonumber\] where $T(0) = T_0$ is the initial temperature of the object, $T_{a}$ is the ambient temperature a and $k>0$ is the constant of proportionality which satisfies the equation

\[\left(\mbox{instantaneous rate of change of $T(t)$ at time $t$}\right) = k \, \left(T(t) - T_{a}\right)\nonumber\]

If we re-examine the situation in Example 6.1.2 with $T_0 = 160$, $T_{a} = 70$, and $k = 0.1$, we get, according to Equation 6.6 , $T(t) = 70 + (160 - 70)e^{-0.1t}$ which reduces to the original formula given. The rate constant $k = 0.1$ indicates the coffee is cooling at a rate equal to $10 \%$ of the difference between the temperature of the coffee and its surroundings. Note in Equation 6.6 that the constant $k$ is positive for both the cooling and warming scenarios. What determines if the function $T(t)$ is increasing or decreasing is if $T_0$ (the initial temperature of the object) is greater than $T_{a}$ (the ambient temperature) or vice-versa, as we see in our next example.

Example 6.5.4.

A $40^{\circ}\mbox{F}$ roast is cooked in a $350^{\circ}\mbox{F}$ oven. After $2$ hours, the temperature of the roast is $125^{\circ}\mbox{F}$.

Assuming the temperature of the roast follows Newton’s Law of Warming, find a formula for the temperature of the roast $T$ as a function of its time in the oven, $t$, in hours.
The roast is done when the internal temperature reaches $165^{\circ}\mbox{F}$. When will the roast be done?
The initial temperature of the roast is $40^{\circ}\mbox{F}$, so $T_0 = 40$. The environment in which we are placing the roast is the $350^{\circ}\mbox{F}$ oven, so $T_{a} = 350$. Newton’s Law of Warming tells us $T(t) = 350 + (40-350)e^{-kt}$, or $T(t) = 350 - 310e^{-kt}$. To determine $k$, we use the fact that after $2$ hours, the roast is $125^{\circ}\mbox{F}$, which means $T(2) = 125$. This gives rise to the equation $350 - 310e^{-2k} = 125$ which yields $k = -\frac{1}{2} \ln \left( \frac{45}{62} \right) \approx 0.1602$. The temperature function is \[T(t) = 350 - 310 e^{\frac{t}{2} \ln \left( \frac{45}{62} \right)} \approx 350- 310 e^{-0.1602 t}.\nonumber\]
To determine when the roast is done, we set $T(t) = 165$. This gives $350- 310 e^{-0.1602 t} = 165$ whose solution is $t = -\frac{1}{0.1602} \ln \left( \frac{37}{62} \right) \approx 3.22$. It takes roughly $3$ hours and $15$ minutes to cook the roast completely.

If we had taken the time to graph $y=T(t)$ in Example 6.5.4 , we would have found the horizontal asymptote to be $y = 350$, which corresponds to the temperature of the oven. We can also arrive at this conclusion by applying a bit of ‘number sense’. As t $\rightarrow \infty,-0.1602 t \approx \text { very big }(-)$ so that $e^{-0.1602 t} \approx \text { very small }(+)$. The larger the value of $t$, the smaller $e^{-0.1602 t}$ becomes so that $T(t) \approx 350-\text { very small }(+)$, which indicates the graph of $y=T(t)$ is approaching its horizontal asymptote $y=350$ from below. Physically, this means the roast will eventually warm up to $350^{\circ}\mbox{F}$. 12 The function $T$ is sometimes called a limited growth model, since the function $T$ remains bounded as $t \rightarrow \infty$. If we apply the principles behind Newton’s Law of Cooling to a biological example, it says the growth rate of a population is directly proportional to how much room the population has to grow. In other words, the more room for expansion, the faster the growth rate. The logistic growth model combines The Law of Uninhibited Growth with limited growth and states that the rate of growth of a population varies jointly with the population itself as well as the room the population has to grow.

Equation 6.7. Logistic Growth

If a population behaves according to the assumptions of logistic growth, the number of organisms $N$ at time $t$ is given by the equation \[N(t) =\dfrac{L}{1 + Ce^{-kLt}},\nonumber\] where $N(0) = N_0$ is the initial population, $L$ is the limiting population, a $C$ is a measure of how much room there is to grow given by \[C = \dfrac{L}{N_0} - 1.\nonumber\] and $k > 0$ is the constant of proportionality which satisfies the equation

\[\left(\mbox{instantaneous rate of change of $N(t)$ at time $t$}\right) = k \, N(t) \left(L - N(t)\right)\nonumber\]

The logistic function is used not only to model the growth of organisms, but is also often used to model the spread of disease and rumors. 13

Example 6.5.5

The number of people $N$, in hundreds, at a local community college who have heard the rumor ‘Carl is afraid of Virginia Woolf’ can be modeled using the logistic equation

\[N(t) = \dfrac{84}{1+2799e^{-t}},\nonumber\]

where $t\geq 0$ is the number of days after April 1, 2009.

Find and interpret $N(0)$.
Find and interpret the end behavior of $N(t)$.
How long until $4200$ people have heard the rumor?
Check your answers to 2 and 3 using your calculator.
We find $N(0) = \frac{84}{1+2799e^{0}} = \frac{84}{2800} = \frac{3}{100}$. Since $N(t)$ measures the number of people who have heard the rumor in hundreds, $N(0)$ corresponds to $3$ people. Since $t=0$ corresponds to April 1, 2009, we may conclude that on that day, $3$ people have heard the rumor. 14
We could simply note that $N(t)$ is written in the form of Equation 6.7 , and identify $L = 84$. However, to see why the answer is $84$, we proceed analytically. Since the domain of $N$ is restricted to $t \geq 0$, the only end behavior of significance is $t \rightarrow \infty$. As we’ve seen before, 15 as $t \rightarrow \infty$, we have $1997 e^{-t} \rightarrow 0^{+}$ and so $N(t) \approx \frac{84}{1+\text { very small }(+)} \approx 84$. Hence, as $t \rightarrow \infty$, $N(t) \rightarrow 84$. This means that as time goes by, the number of people who will have heard the rumor approaches $8400$.
To find how long it takes until $4200$ people have heard the rumor, we set $N(t) = 42$. Solving $\frac{84}{1+2799e^{-t}} = 42$ gives $t = \ln(2799) \approx 7.937$. It takes around $8$ days until $4200$ people have heard the rumor.

$Screen Shot 2022-04-18 at 7.12.22 PM.png$

If we take the time to analyze the graph of $y=N(x)$ above, we can see graphically how logistic growth combines features of uninhibited and limited growth. The curve seems to rise steeply, then at some point, begins to level off. The point at which this happens is called an inflection point or is sometimes called the ‘point of diminishing returns’. At this point, even though the function is still increasing, the rate at which it does so begins to decline. It turns out the point of diminishing returns always occurs at half the limiting population. (In our case, when $y=42$.) While these concepts are more precisely quantified using Calculus, below are two views of the graph of $y=N(x)$, one on the interval $[0,8]$, the other on $[8,15]$. The former looks strikingly like uninhibited growth; the latter like limited growth.

$Screen Shot 2022-04-18 at 7.19.06 PM.png$

6.5.2 Applications of Logarithms

Just as many physical phenomena can be modeled by exponential functions, the same is true of logarithmic functions. In Exercises 75 , 76 and 77 of Section 6.1 , we showed that logarithms are useful in measuring the intensities of earthquakes (the Richter scale), sound (decibels) and acids and bases (pH). We now present yet a different use of the a basic logarithm function, password strength .

Example 6.5.6

The information entropy $H$, in bits, of a randomly generated password consisting of $L$ characters is given by $H = L \log_{2}(N)$, where $N$ is the number of possible symbols for each character in the password. In general, the higher the entropy, the stronger the password.

If a $7$ character case-sensitive 16 password is comprised of letters and numbers only, find the associated information entropy.
How many possible symbol options per character is required to produce a $7$ character password with an information entropy of $50$ bits?
There are $26$ letters in the alphabet, $52$ if upper and lower case letters are counted as different. There are $10$ digits ($0$ through $9$) for a total of $N=62$ symbols. Since the password is to be $7$ characters long, $L = 7$. Thus, $H = 7 \log_{2}(62) = \frac{7 \ln(62)}{\ln(2)} \approx 41.68$.
We have $L = 7$ and $H=50$ and we need to find $N$. Solving the equation $50 = 7 \log_{2}(N)$ gives $N = 2^{50/7} \approx 141.323$, so we would need $142$ different symbols to choose from. 17

Chemical systems known as buffer solutions have the ability to adjust to small changes in acidity to maintain a range of pH values. Buffer solutions have a wide variety of applications from maintaining a healthy fish tank to regulating the pH levels in blood. Our next example shows how the pH in a buffer solution is a little more complicated than the pH we first encountered in Exercise 77 in Section 6.1 .

Example 6.5.7

Blood is a buffer solution. When carbon dioxide is absorbed into the bloodstream it produces carbonic acid and lowers the pH. The body compensates by producing bicarbonate, a weak base to partially neutralize the acid. The equation 18 which models blood pH in this situation is $\mbox{pH} = 6.1 + \log\left(\frac{800}{x} \right)$, where $x$ is the partial pressure of carbon dioxide in arterial blood, measured in torr. Find the partial pressure of carbon dioxide in arterial blood if the pH is $7.4$.

We set $\mbox{pH} = 7.4$ and get $7.4 = 6.1 + \log\left(\frac{800}{x} \right)$, or $\log\left(\frac{800}{x} \right) = 1.3$. Solving, we find $x = \frac{800}{10^{1.3}} \approx 40.09$. Hence, the partial pressure of carbon dioxide in the blood is about $40$ torr.

Another place logarithms are used is in data analysis. Suppose, for instance, we wish to model the spread of influenza A (H1N1), the so-called ‘Swine Flu’. Below is data taken from the World Health Organization (WHO) where $t$ represents the number of days since April 28, 2009, and $N$ represents the number of confirmed cases of H1N1 virus worldwide.

\[\begin{array}{|c||c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline t & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 \\ \hline N & 148 & 257 & 367 & 658 & 898 & 1085 & 1490 & 1893 & 2371 & 2500 & 3440 & 4379 & 4694 \\ \hline \end{array}\nonumber\]

\[\begin{array}{|c||c||c|c|c|c|c|c|} \hline t & 14 & 15 & 16 & 17 & 18 & 19& 20 \\ \hline N & 5251 & 5728 & 6497 & 7520 & 8451 & 8480 & 8829 \\ \hline \end{array}\nonumber\]

Making a scatter plot of the data treating $t$ as the independent variable and $N$ as the dependent variable gives

$Screen Shot 2022-04-18 at 7.46.14 PM.png$

Which models are suggested by the shape of the data? Thinking back Section 2.5 , we try a Quadratic Regression, with pretty good results.

$Screen Shot 2022-04-18 at 7.49.10 PM.png$

However, is there any scientific reason for the data to be quadratic? Are there other models which fit the data equally well, or better? Scientists often use logarithms in an attempt to ‘linearize’ data sets - in other words, transform the data sets to produce ones which result in straight lines. To see how this could work, suppose we guessed the relationship between $N$ and $t$ was some kind of power function, not necessarily quadratic, say $N = B t^{A}$. To try to determine the $A$ and $B$, we can take the natural log of both sides and get $\ln(N) = \ln\left(B t^{A}\right)$. Using properties of logs to expand the right hand side of this equation, we get $\ln(N) = A \ln(t) + \ln(B)$. If we set $X = \ln(t)$ and $Y = \ln(N)$, this equation becomes $Y = AX + \ln(B)$. In other words, we have a line with slope $A$ and $Y$-intercept $\ln(B)$. So, instead of plotting $N$ versus $t$, we plot $\ln(N)$ versus $\ln(t)$.

\[\begin{array}{|c||c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline \ln(t) & 0 & 0.693 & 1.099 & 1.386& 1.609 & 1.792 & 1.946 & 2.079 & 2.197 & 2.302 & 2.398 & 2.485 & 2.565 \\ \hline \ln(N) & 4.997 & 5.549 & 5.905 & 6.489 & 6.800 & 6.989 & 7.306 & 7.546 & 7.771 & 7.824 & 8.143 & 8.385 & 8.454 \\ \hline \end{array}\nonumber\]

\[\begin{array}{|c||c||c|c|c|c|c|c|} \hline \ln(t) & 2.639 & 2.708 & 2.773 & 2.833 & 2.890 & 2.944 & 2.996 \\ \hline \ln(N) & 8.566 & 8.653 & 8.779 & 8.925 & 9.042 & 9.045 & 9.086 \\ \hline \end{array}\nonumber\]

Running a linear regression on the data gives

$Screen Shot 2022-04-18 at 7.50.26 PM.png$

The slope of the regression line is $a \approx 1.512$ which corresponds to our exponent $A$. The $y$-intercept $b \approx 4.513$ corresponds to $\ln(B)$, so that $B \approx 91.201$. Hence, we get the model $N = 91.201 t^{1.512}$, something from Section 5.3 . Of course, the calculator has a built-in ‘Power Regression’ feature. If we apply this to our original data set, we get the same model we arrived at before. 19

$Screen Shot 2022-04-18 at 7.52.22 PM.png$

This is all well and good, but the quadratic model appears to fit the data better, and we’ve yet to mention any scientific principle which would lead us to believe the actual spread of the flu follows any kind of power function at all. If we are to attack this data from a scientific perspective, it does seem to make sense that, at least in the early stages of the outbreak, the more people who have the flu, the faster it will spread, which leads us to proposing an uninhibited growth model. If we assume $N = B e^{At}$ then, taking logs as before, we get $\ln(N) = At + \ln(B)$. If we set $X = t$ and $Y = \ln(N)$, then, once again, we get $Y = AX + \ln(B)$, a line with slope $A$ and $Y$-intercept $\ln(B)$. Plotting $\ln(N)$ versus $t$ gives the following linear regression.

$Screen Shot 2022-04-18 at 7.53.05 PM.png$

We see the slope is $a \approx 0.202$ and which corresponds to $A$ in our model, and the $y$-intercept is $b \approx 5.596$ which corresponds to $\ln(B)$. We get $B \approx 269.414$, so that our model is $N = 269.414e^{0.202t}$. Of course, the calculator has a built-in ‘Exponential Regression’ feature which produces what appears to be a different model $N = 269.414 (1.22333419)^{t}$. Using properties of exponents, we write $e^{0.202t} = \left(e^{0.202}\right)^t \approx (1.223848)^{t}$, which, had we carried more decimal places, would have matched the base of the calculator model exactly.

$Screen Shot 2022-04-18 at 7.55.48 PM.png$

The exponential model didn’t fit the data as well as the quadratic or power function model, but it stands to reason that, perhaps, the spread of the flu is not unlike that of the spread of a rumor and that a logistic model can be used to model the data. The calculator does have a ‘Logistic Regression’ feature, and using it produces the model $N = \frac{10739.147}{1 + 42.416 e^{0.268 t}}$.

$Screen Shot 2022-04-18 at 7.56.22 PM.png$

This appears to be an excellent fit, but there is no friendly coefficient of determination, $R^2$, by which to judge this numerically. There are good reasons for this, but they are far beyond the scope of the text. Which of the models, quadratic, power, exponential, or logistic is the ‘best model’? If by ‘best’ we mean ‘fits closest to the data,’ then the quadratic and logistic models are arguably the winners with the power function model a close second. However, if we think about the science behind the spread of the flu, the logistic model gets an edge. For one thing, it takes into account that only a finite number of people will ever get the flu (according to our model, $10,\!739$), whereas the quadratic model predicts no limit to the number of cases. As we have stated several times before in the text, mathematical models, regardless of their sophistication, are just that: models, and they all have their limitations. 20

6.5.3. Exercises

For each of the scenarios given in Exercises 1 - 6,

Find the amount $A$ in the account as a function of the term of the investment $t$ in years.
Determine how much is in the account after $5$ years, $10$ years, $30$ years and $35$ years. Round your answers to the nearest cent.
Determine how long will it take for the initial investment to double. Round your answer to the nearest year.
Find and interpret the average rate of change of the amount in the account from the end of the fourth year to the end of the fifth year, and from the end of the thirty-fourth year to the end of the thirty-fifth year. Round your answer to two decimal places.
$\$500$ is invested in an account which offers $0.75 \%$, compounded monthly.
$\$500$ is invested in an account which offers $0.75 \%$, compounded continuously.
$\$1000$ is invested in an account which offers $1.25 \%$, compounded monthly.
$\$1000$ is invested in an account which offers $1.25 \%$, compounded continuously.
$\$5000$ is invested in an account which offers $2.125 \%$, compounded monthly.
$\$5000$ is invested in an account which offers $2.125 \%$, compounded continuously.
Look back at your answers to Exercises 1 - 6. What can be said about the difference between monthly compounding and continuously compounding the interest in those situations? With the help of your classmates, discuss scenarios where the difference between monthly and continuously compounded interest would be more dramatic. Try varying the interest rate, the term of the investment and the principal. Use computations to support your answer.
How much money needs to be invested now to obtain $\$2000$ in 3 years if the interest rate in a savings account is $0.25 \%$, compounded continuously? Round your answer to the nearest cent.
How much money needs to be invested now to obtain $\$5000$ in 10 years if the interest rate in a CD is $2.25 \%$, compounded monthly? Round your answer to the nearest cent.
If $P = 2000$ what is $A(8)$?
Solve the equation $A(t) = 4000$ for $t$.
What principal $P$ should be invested so that the account balance is $2000 is three years?
What principal $P$ should be invested so that the account balance is $2000 in three years?
The Annual Percentage Yield is the interest rate that returns the same amount of interest after one year as the compound interest does. With the help of your classmates, compute the APY for this investment.
A finance company offers a promotion on $\$5000$ loans. The borrower does not have to make any payments for the first three years, however interest will continue to be charged to the loan at $29.9 \%$ compounded continuously. What amount will be due at the end of the three year period, assuming no payments are made? If the promotion is extended an additional three years, and no payments are made, what amount would be due?
Use Equation 6.2 to show that the time it takes for an investment to double in value does depend on the principal $P$, but rather, depends only on the APR and the number of compoundings per year. Let $n = 12$ and with the help of your classmates compute the doubling time for a variety of rates $r$. Then look up the Rule of 72 and compare your answers to what that rule says. If you’re really interested 21 in Financial Mathematics, you could also compare and contrast the Rule of 72 with the Rule of 70 and the Rule of 69.

In Exercises 14 - 18, we list some radioactive isotopes and their associated half-lives. Assume that each decays according to the formula $A(t)=A_{0} e^{k t}$ where $A_{0}$ is the initial amount of the material and $k$ is the decay constant. For each isotope:

Find the decay constant $k$. Round your answer to four decimal places.
Find a function which gives the amount of isotope $A$ which remains after time $t$. (Keep the units of $A$ and $t$ the same as the given data.)
Determine how long it takes for $90 \%$ of the material to decay. Round your answer to two decimal places. (HINT: If $90 \%$ of the material decays, how much is left?)
Cobalt 60, used in food irradiation, initial amount 50 grams, half-life of $5.27$ years.
Phosphorus 32, used in agriculture, initial amount 2 milligrams, half-life $14$ days.
Chromium 51, used to track red blood cells, initial amount 75 milligrams, half-life $27.7$ days.
Americium 241, used in smoke detectors, initial amount 0.29 micrograms, half-life $432.7$ years.
Uranium 235, used for nuclear power, initial amount $1$ kg grams, half-life $704$ million years.
With the help of your classmates, show that the time it takes for $90 \%$ of each isotope listed in Exercises 14 - 18 to decay does not depend on the initial amount of the substance, but rather, on only the decay constant $k$. Find a formula, in terms of $k$ only, to determine how long it takes for $90 \%$ of a radioactive isotope to decay.
In Example 6.1.1 in Section 6.1 , the exponential function $V(x) = 25 \left(\frac{4}{5}\right)^{x}$ was used to model the value of a car over time. Use the properties of logs and/or exponents to rewrite the model in the form $V(t) = 25e^{kt}$.
Find and interpret $G(0)$.
According to the model, what should have been the GDP in 2007? In 2010? (According to the US Department of Commerce , the 2007 GDP was $\$14,369.1$ billion and the 2010 GDP was $\$14,657.8$ billion.)
What was the diameter of the tumor when it was originally detected?
How long until the diameter of the tumor doubles?
Find the growth constant $k$. Round your answer to four decimal places.
Find a function which gives the number of bacteria $N(t)$ after $t$ minutes.
How long until there are 9000 bacteria? Round your answer to the nearest minute.
Find a function which gives the number of yeast (in millions) per cc $N(t)$ after $t$ hours.
What is the doubling time for this strain of yeast?
The Law of Uninhibited Growth also applies to situations where an animal is re-introduced into a suitable environment. Such a case is the reintroduction of wolves to Yellowstone National Park. According to the National Park Service , the wolf population in Yellowstone National Park was 52 in 1996 and 118 in 1999. Using these data, find a function of the form $N(t)=N_{0} e^{k t}$ which models the number of wolves $t$ years after 1996. (Use $t = 0$ to represent the year 1996. Also, round your value of $k$ to four decimal places.) According to the model, how many wolves were in Yellowstone in 2002? (The recorded number is 272.)
During the early years of a community, it is not uncommon for the population to grow according to the Law of Uninhibited Growth. According to the Painesville Wikipedia entry, in 1860, the Village of Painesville had a population of 2649. In 1920, the population was 7272. Use these two data points to fit a model of the form $N(t)=N_{0} e^{k t}$ were $N(t)$ is the number of Painesville Residents $t$ years after 1860. (Use $t = 0$ to represent the year 1860. Also, round the value of $k$ to four decimal places.) According to this model, what was the population of Painesville in 2010? (The 2010 census gave the population as 19,563) What could be some causes for such a vast discrepancy? For more on this, see Exercise 37.
Find and interpret $P(0)$.
Find the population of Sasquatch in Bigfoot county in 2013. Round your answer to the nearest Sasquatch.
When will the population of Sasquatch in Bigfoot county reach 60? Round your answer to the nearest year.
Find and interpret the end behavior of the graph of $y = P(t)$. Check your answer using a graphing utility.
Use Equation 6.5 to express the amount of Carbon-14 left from an initial $N$ milligrams as a function of time $t$ in years.
What percentage of the original amount of Carbon-14 is left after 20,000 years?
If an old wooden tool is found in a cave and the amount of Carbon-14 present in it is estimated to be only 42% of the original amount, approximately how old is the tool?
Radiocarbon dating is not as easy as these exercises might lead you to believe. With the help of your classmates, research radiocarbon dating and discuss why our model is somewhat over-simplified.
Carbon-14 cannot be used to date inorganic material such as rocks, but there are many other methods of radiometric dating which estimate the age of rocks. One of them, Rubidium-Strontium dating, uses Rubidium-87 which decays to Strontium-87 with a half-life of 50 billion years. Use Equation 6.5 to express the amount of Rubidium-87 left from an initial 2.3 micrograms as a function of time $t$ in billions of years. Research this and other radiometric techniques and discuss the margins of error for various methods with your classmates.
Use Equation 6.5 to show that $k = -\dfrac{\ln(2)}{h}$ where $h$ is the half-life of the radioactive isotope.
Express the temperature $T$ (in $^{\circ}$F) as a function of time $t$ (in minutes).
Find the time at which the roast would have dropped to $140^{\circ}$F had it not been carved and eaten.

\[y(x) = \frac{1}{4} x^2-\frac{1}{4} \ln(x)-\frac{1}{4}\nonumber\]

Use your calculator to graph this path for $x > 0$. Describe the behavior of $y$ as $x \rightarrow 0^{+}$ and interpret this physically.

The current $i$ measured in amps in a certain electronic circuit with a constant impressed voltage of 120 volts is given by $i(t) = 2 - 2e^{-10t}$ where $t \geq 0$ is the number of seconds after the circuit is switched on. Determine the value of $i$ as $t \rightarrow \infty$. (This is called the steady state current.)

\[i(t) = \left\{ \begin{array}{rcl} 2 - 2e^{-10t} & \mbox{if} & 0 \leq t < 30 \\ [6pt] \left(2 - 2e^{-300}\right) e^{-10t+300} & \mbox{if} & t \geq 30 \end{array} \right.\nonumber\]

With the help of your calculator, graph $y = i(t)$ and discuss with your classmates the physical significance of the two parts of the graph $0 \leq t < 30$ and $t \geq 30$.

In Exercise 6a in Section 2.5 , we stated that the cable of a suspension bridge formed a parabola but that a free hanging cable did not. A free hanging cable forms a and its basic shape is given by $y = \frac{1}{2}\left(e^{x} + e^{-x}\right)$. Use your calculator to graph this function. What are its domain and range? What is its end behavior? Is it invertible? How do you think it is related to the function given in Exercise 47 in Section 6.3 and the one given in the answer to Exercise 38 in Section 6.4 ? When flipped upside down, the catenary makes an arch. The Gateway Arch in St. Louis, Missouri has the shape \[y = 757.7 - \frac{127.7}{2}\left(e^{\frac{x}{127.7}} + e^{-\frac{x}{127.7}}\right)\nonumber\] where $x$ and $y$ are measured in feet and $-315 \leq x \leq 315$. Find the highest point on the arch.

$\begin{array}{|l|r|r|r|r|r|r|r|r|r|r|} \hline \text { Year } x & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & & 9 \\ \hline \text { Number of } & & & & & & & & & & \\ \text { Cats } N(x) & 12 & 66 & 382 & 2201 & 12680 & 73041 & 420715 & 2423316 & 13968290 & 80399780 \\ \hline \end{array}$

Use a graphing utility to perform an exponential regression on the data from 1860 through 1920 only, letting $t = 0$ represent the year 1860 as before. How does this calculator model compare with the model you found in Exercise 26? Use the calculator’s exponential model to predict the population in 2010. (The 2010 census gave the population as 19,563)
The logistic model fit to all of the given data points for the population of Painesville $t$ years after 1860 (again, using $t = 0$ as 1860) is \[P(t) = \dfrac{18691}{1+9.8505e^{-0.03617t}}\nonumber\] According to this model, what should the population of Painesville have been in 2010? (The 2010 census gave the population as 19,563.) What is the population limit of Painesville?
Use your calculator to fit a logistic model to these data, using $x = 0$ to represent the year 1860.
Graph these data and your logistic function on your calculator to judge the reasonableness of the fit.
Use this model to estimate the population of Lake County in 2010. (The 2010 census gave the population to be 230,041.)
According to your model, what is the population limit of Lake County, Ohio?

With the help of your classmates, find a model for this data.

With the help of your classmates, find a model for this data. Unlike most of the phenomena we have studied in this section, there is no single differential equation which governs the enrollment growth. Thus there is no scientific reason to rely on a logistic function even though the data plot may lead us to that model. What are some factors which influence enrollment at a community college and how can you take those into account mathematically?

With the help of your classmates, find a model for this data and make a prediction for the Opening Day enrollment as well as the Day 15 enrollment. (WARNING: The registration period for 2009 was one week shorter than it was in 2008 so Opening Day would be $x = 21$ and Day 15 is $x = 23$.)

6.5.4. Answers

$A(t) = 500\left(1 + \frac{0.0075}{12}\right)^{12t}$
$A(5) \approx \$ 519.10$, $A(10) \approx \$ 538.93$, $A(30) \approx \$ 626.12$, $A(35) \approx \$ 650.03$
It will take approximately $92$ years for the investment to double.
The average rate of change from the end of the fourth year to the end of the fifth year is approximately $3.88$. This means that the investment is growing at an average rate of $\$3.88$ per year at this point. The average rate of change from the end of the thirty-fourth year to the end of the thirty-fifth year is approximately $4.85$. This means that the investment is growing at an average rate of $\$4.85$ per year at this point.
$A(t) = 500e^{0.0075t}$
$A(5) \approx \$ 519.11$, $A(10) \approx \$ 538.94$, $A(30) \approx \$ 626.16$, $A(35) \approx \$ 650.09$
The average rate of change from the end of the fourth year to the end of the fifth year is approximately $3.88$. This means that the investment is growing at an average rate of $\$3.88$ per year at this point. The average rate of change from the end of the thirty-fourth year to the end of the thirty-fifth year is approximately $4.86$. This means that the investment is growing at an average rate of $\$4.86$ per year at this point.
$A(t) = 1000\left(1 + \frac{0.0125}{12}\right)^{12t}$
$A(5) \approx \$ 1064.46$, $A(10) \approx \$ 1133.07$, $A(30) \approx \$ 1454.71$, $A(35) \approx \$ 1548.48$
It will take approximately $55$ years for the investment to double.
The average rate of change from the end of the fourth year to the end of the fifth year is approximately $13.22$. This means that the investment is growing at an average rate of $\$13.22$ per year at this point. The average rate of change from the end of the thirty-fourth year to the end of the thirty-fifth year is approximately $19.23$. This means that the investment is growing at an average rate of $\$19.23$ per year at this point.
$A(t) = 1000e^{0.0125t}$
$A(5) \approx \$ 1064.49$, $A(10) \approx \$ 1133.15$, $A(30) \approx \$ 1454.99$, $A(35) \approx \$ 1548.83$
The average rate of change from the end of the fourth year to the end of the fifth year is approximately $13.22$. This means that the investment is growing at an average rate of $\$13.22$ per year at this point. The average rate of change from the end of the thirty-fourth year to the end of the thirty-fifth year is approximately $19.24$. This means that the investment is growing at an average rate of $\$19.24$ per year at this point.
$A(t) = 5000\left(1 + \frac{0.02125}{12}\right)^{12t}$
$A(5) \approx \$ 5559.98$, $A(10) \approx \$ 6182.67$, $A(30) \approx \$ 9453.40$, $A(35) \approx \$ 10512.13$
It will take approximately $33$ years for the investment to double.
The average rate of change from the end of the fourth year to the end of the fifth year is approximately $116.80$. This means that the investment is growing at an average rate of $\$116.80$ per year at this point. The average rate of change from the end of the thirty-fourth year to the end of the thirty-fifth year is approximately $220.83$. This means that the investment is growing at an average rate of $\$220.83$ per year at this point.
$A(t) = 5000e^{0.02125t}$
$A(5) \approx \$ 5560.50$, $A(10) \approx \$ 6183.83$, $A(30) \approx \$ 9458.73$, $A(35) \approx \$ 10519.05$
The average rate of change from the end of the fourth year to the end of the fifth year is approximately $116.91$. This means that the investment is growing at an average rate of $\$116.91$ per year at this point. The average rate of change from the end of the thirty-fourth year to the end of the thirty-fifth year is approximately $221.17$. This means that the investment is growing at an average rate of $\$221.17$ per year at this point.
$P = \frac{2000}{e^{0.0025 \cdot 3}} \approx \$ 1985.06$
$P = \frac{5000}{\left(1 + \frac{0.0225}{12}\right)^{12 \cdot 10}} \approx \$ 3993.42$
$A(8) = 2000\left(1 + \frac{0.0025}{12}\right)^{12 \cdot 8} \approx \$2040.40$
$t = \dfrac{\ln(2)}{12 \ln\left(1 + \frac{0.0025}{12}\right)} \approx 277.29$ years
$P = \dfrac{2000}{\left(1 + \frac{0.0025}{12}\right)^{36}} \approx \$1985.06$
$A(8) = 2000\left(1 + \frac{0.0225}{12}\right)^{12 \cdot 8} \approx \$2394.03$
$t = \dfrac{\ln(2)}{12 \ln\left(1 + \frac{0.0225}{12}\right)} \approx 30.83$ years
$P = \dfrac{2000}{\left(1 + \frac{0.0225}{12}\right)^{36}} \approx \$1869.57$
$\left(1 + \frac{0.0225}{12}\right)^{12} \approx 1.0227$ so the APY is 2.27%
$A(3) = 5000e^{0.299 \cdot 3} \approx \$12,226.18$, $A(6) = 5000e^{0.299 \cdot 6} \approx \$30,067.29$
$k = \frac{\ln(1/2)}{5.27} \approx -0.1315$
$A(t) = 50e^{-0.1315t}$
$t = \frac{\ln(0.1)}{-0.1315} \approx 17.51$ years.
$k = \frac{\ln(1/2)}{14} \approx -0.0495$
$A(t) = 2e^{-0.0495t}$
$t = \frac{\ln(0.1)}{-0.0495} \approx 46.52$ days.
$k = \frac{\ln(1/2)}{27.7} \approx -0.0250$
$A(t) = 75e^{-0.0250t}$
$t = \frac{\ln(0.1)}{-0.025} \approx 92.10$ days.
$k = \frac{\ln(1/2)}{432.7} \approx -0.0016$
$A(t) = 0.29e^{-0.0016t}$
$t = \frac{\ln(0.1)}{-0.0016} \approx 1439.11$ years.
$k = \frac{\ln(1/2)}{704} \approx -0.0010$
$A(t) = e^{-0.0010t}$
$t = \frac{\ln(0.1)}{-0.0010} \approx 2302.58$ million years, or $2.30$ billion years.
$t = \frac{\ln(0.1)}{k} = -\frac{\ln(10)}{k}$
$V(t) = 25e^{\ln\left(\frac{4}{5}\right)t} \approx 25e^{-0.22314355t}$
$G(0) = 9743.77$ This means that the GDP of the US in 2000 was $\$9743.77$ billion dollars.
$G(7) = 13963.24$ and $G(10) = 16291.25$, so the model predicted a GDP of $\$ 13,963.24$ billion in 2007 and $\$ 16,291.25$ billion in 2010.
$D(0) = 15$, so the tumor was 15 millimeters in diameter when it was first detected.
$t = \frac{\ln(2)}{0.0277} \approx 25$ days.
$k = \frac{\ln(2)}{20} \approx 0.0346$
$N(t) = 1000e^{0.0346 t}$
$t = \frac{\ln(9)}{0.0346} \approx 63$ minutes
$k = \frac{1}{2}\frac{\ln(6)}{2.5} \approx 0.4377$
$N(t) = 2.5e^{0.4377 t}$
$t = \frac{\ln(2)}{0.4377} \approx 1.58$ hours
$N_{0}=52, k=\frac{1}{3} \ln \left(\frac{118}{52}\right) \approx 0.2731, N(t)=52 e^{0.2731 t} \cdot N(6) \approx 268$.
$N_{0}=2649, k=\frac{1}{60} \ln \left(\frac{7272}{2649}\right) \approx 0.0168, N(t)=2649 e^{0.0168 t} \cdot N(150) \approx 32923$, so the population of Painesville in 2010 based on this model would have been 32,923.
$P(0) = \frac{120}{4.167} \approx 29$. There are 29 Sasquatch in Bigfoot County in 2010.
$P(3) = \frac{120}{1+3.167e^{-0.05(3)}} \approx 32$ Sasquatch.
$t = 20 \ln(3.167) \approx 23$ years.
As $t \rightarrow \infty$, $P(t) \rightarrow 120$. As time goes by, the Sasquatch Population in Bigfoot County will approach 120. Graphically, $y = P(x)$ has a horizontal asymptote $y=120$.
$A(t) = Ne^{-\left(\frac{\ln(2)}{5730}\right)t} \approx Ne^{-0.00012097t}$
$A(20000) \approx 0.088978 \cdot N$ so about 8.9% remains
$t \approx \dfrac{\ln(.42)}{-0.00012097} \approx 7171$ years old
$A(t) = 2.3e^{-0.0138629t}$
$T(t) = 75 + 105e^{-0.005005t}$
The roast would have cooled to $140^{\circ}$F in about 95 minutes.

$Screen Shot 2022-04-19 at 6.33.34 PM.png$

The steady state current is 2 amps.

$N(x) = 2.02869(5.74879)^{x} = 2.02869e^{1.74899x}$ with $r^{2} \approx 0.999995$. This is also an excellent fit and corresponds to our linearized model because $\ln(2.02869) \approx 0.70739$.

The calculator gives: $y = 2895.06 (1.0147)^{x}$. Graphing this along with our answer from Exercise 26 over the interval $[0,60]$ shows that they are pretty close. From this model, $y(150) \approx 25840$ which once again overshoots the actual data value.
$P(150) \approx 18717$, so this model predicts 17,914 people in Painesville in 2010, a more conservative number than was recorded in the 2010 census. As $t \rightarrow \infty$, $P(t) \rightarrow 18691$. So the limiting population of Painesville based on this model is 18,691 people.
$y = \dfrac{242526}{1+874.62e^{-0.07113x}}$, where $x$ is the number of years since 1860.

$Screen Shot 2022-04-21 at 4.08.08 PM.png$

$y(140) \approx 232889$, so this model predicts 232,889 people in Lake County in 2010.
As $x \rightarrow \infty$, $y \rightarrow 242526$, so the limiting population of Lake County based on this model is 242,526 people.

1 How generous of the

2 Some restrictions may apply.

3 Actually, the final balance should be $105.0625.

4 Using this convention, simple interest after one year is the same as compounding the interest only once.

5 See Definition 2.3 in section 2.1 .

6 In fact, the rate of increase of the amount in the account is exponential as well. This is the quality that really defines exponential functions and we refer the reader to a course in Calculus.

7 Once you’ve had a semester of Calculus, you’ll be able to fully appreciate this very lame pun.

8 Or define, depending on your point of view.

9 The average rate of change of a function over an interval was first introduced in Section 2.1 . Instantaneous rates of change are the business of Calculus, as is mentioned on Page 161.

10 The time it takes for half of the substance to decay.

11 The Second Law of Thermodynamics states that heat can spontaneously flow from a hotter object to a colder one, but not the other way around. Thus, the coffee could not continue to release heat into the air so as to cool below room temperature.

12 at which point it would be more toast than roast.

13 Which can be just as damaging as diseases.

14 Or, more likely, three people started the rumor. I’d wager Jeff, Jamie, and Jason started it. So much for telling your best friends something in confidence!

15 See, for example, Example 6.1.2 .

16 That is, upper and lower case letters are treated as different characters.

17 Since there are only 94 distinct ASCII keyboard characters, to achieve this strength, the number of characters in the password should be increase.

18 Derived from the Henderson-Hasselbalch Equation . See Exercise 43 in Section 6.2 . Hasselbalch himself was studying carbon dioxide dissolving in blood - a process called metabolic acidosis .

19 Critics may question why the authors of the book have chosen to even discuss linearization of data when the calculator has a Power Regression built-in and ready to go. Our response: talk to your science faculty

20 Speaking of limitations, as of June 3, 2009, there were 19,273 confirmed cases of influenza A (H1N1). This is well above our prediction of 10,739. Each time a new report is issued, the data set increases and the model must be recalculated. We leave this recalculation to the reader.

21 Awesome pun!

22 This roast was enjoyed by Jeff and his family on June 10, 2009. This is real data, folks!

23 The authors thank Dr. Wendy Marley and her staff for this data and Dr. Marcia Ballinger for the permission to use it in this problem.

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