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A logarithm is the inverse of the exponential function . Specifically, a logarithm is the power to which a number (the base) must be raised to produce a given number.

For example, \(\log_2 64 = 6,\) because \( 2^6 = 64.\) In general, we have the following definition:

\( z \) is the base-\(x\) logarithm of \(y\) if and only if \( x^z = y \). In typical notation \[ \log_x y = z \iff x^z = y.\]

Properties of Logarithms - Basic

Worked examples using properties, properties of logarithms - intermediate, problem solving - basic, problem solving - intermediate, problem solving - advanced, applications.

First, we must know the basic structure of a logarithm \((\)abbreviated \(\log\) for convenience\().\) \(\log_a{b}=c\) can be rewritten as \(a^c=b,\) where \(a\) is called the base , \(c\) the exponent , and \(b\) the argument . Also, \(\log\) without a base is shorthand for the common \(\log\) of base \(10.\) Now that we know this, we can manipulate logs:

Other properties can be derived from these basic ones, especially when noting that these properties are inversable.

Simplify \(\log_2 \left(\dfrac{32}{9}\right)^2\) as much as possible. Try to follow the steps and identify what properties were used: \[\begin{align} \log_2 \left(\dfrac{32}{9}\right)^2 &=2 \cdot \log_2 \left(\frac{32}{9}\right)\\ &=2 \cdot ( \log_2 32 - \log_2 9)\\ &=2 \cdot \left( \log_2 2^5 -\log_2 3^2\right)\\ &=2 \cdot ( 5 \cdot \log_2 2 - 2 \cdot \log_2 3)\\ &=2 \cdot ( 5 \cdot 1 -2 \cdot \log_2 3)\\ &=10- 4 \log_2 3. \end{align}\] Note: \(\log_2 3\) can't be simplified further. Line 1 used the second property, line 2 put thingies into exponential form, line 3 used the third property, and lines 4 and 5 did basic simplification. \(_\square\)

Simplify \[\displaystyle{2\log_4{\sqrt{5}}+\frac{1}{2}\log_2{625}-\log_2{\frac{1}{5}}}.\] Again, try to follow the steps of the solution: \[\begin{align} 2\log_{(2^2)}{\big(5^\frac{1}{2}\big)}+\frac{1}{2}\log_2{\big(5^4\big)}-\log_2{\big(5^{-1}\big)} &=2\frac{\log 5^{\frac{1}{2}}}{\log 2^2}+\frac{1}{2}(4)(\log_2{5})+\log_2{5}\\ &=2\frac{\frac{1}{2}\log 5}{2\log 2}+2\log_2{5}+\log_2{5}\\ &=\frac{1}{2}\frac{\log 5}{\log 2}+3\log_2{5}\\ &=\frac{1}{2}\log_2{5}+3\log_2{5}\\ &=\frac{7}{2}\log_2{5}. \end{align}\] The first line shows that it is (usually) best to convert numbers so that they are integers to a power. Note that lines 4 reverses the process of the fourth property. \(_\square\)

\[ \] \(1.~\log _{ a }{ a } =1\)

Find the value of \(\log _{ 4 }{ 4 }.\) Using the property \(\log _{ a }{ a }=1,\) we get \(\log_{ 4 }{ 4 } =1. \ _\square\)

\[ \] \(2.~\log _{ a }{ (b^c) } =c\log _{ a }{ b } \)

Find the value of \(\log _{ 2 }{ 16 }.\) We have \[\begin{align} \log _{ 2 }{ 16 } &= \log _{ 2 }{ { 2 }^{ 4 } } &&\qquad \big(16={ 2 }^{ 4 }\big)\\ &=4\log _{ 2 }{ 2 } &&\qquad \big(\log { { a }^{ b } } = b\log { a } \big)\\ &= 4. \ _\square &&\qquad (\text{by property 1}) \end{align}\]

\[ \] \(3.~\log _{ a }{ (b \times c) } = \log _{ a }{ b }+ \log _{ a }{ c } \)

Find the value of \(\log { 90 }\) assuming \(\log { 3 } =0.47\). We have \[\begin{align} \log { 90 } &= \log { (9\times 10) } &&\qquad (90= 9 \times 10)\\ &=\log { 9 } + \log { 10 } &&\qquad \big(\log _{ a }{ (b\times c) } =\log _{ a }{ b } +\log _{ a }{ c } \big)\\ &=2\log { 3 } +1 &&\qquad \text{(by properties 2 and 1)}\\ &=2\times 0.47+1\\ &=0.94+1\\ &=1.94. \ _\square \end{align}\]

\[ \] \(4.~\displaystyle{\log _{ a }{ \frac { b }{ c } } = \log _{ a }{ b } - \log _{ a }{ c }}\)

Evaluate \(\log { 0.27 }\) assuming \(\log { 3 } =0.47\). We have \[\begin{align} \log { 0.27 } &= \log { \frac { 27 }{ 100 } } \\ &= \log { 27 } - \log { 100 } \\ &=3\log{ 3 } - 2\\ &=1.41 - 2\\ &=-0.59. \ _\square \end{align}\]

\[ \] \(5.\) \(\displaystyle{\log_{ a }{ b } = \frac { \log_{ c }{ b } }{ \log_{ c }{ a } }} \)

Find the value of \(\log_{ 32 }{ 2 }\). We have \[\begin{align} \log_{ 32 }{ 2 } &=\frac { \log_{ 2 }{ 2 } }{ \log_{ 2 }{ 32 } } \\ &=\frac { 1 }{ 5\log_{ 2 }{ 2 } } \\ &=\frac { 1 }{ 5 }\\ &={ 0.2 }. \ _\square \end{align} \]

What is the value of \( \log_3 15 + \log_3 81 - \log_3 5 ?\) Using the properties of logarithms, we can rewrite the given expression as follows: \[ \begin{align} \log_3 15 + \log_3 81 - \log_3 5 &= \log_3 15 - \log_3 5 + \log_3 3^4 \\ &= \log_3 \frac{15}{5} + \log_3 3^4 \\ &= \log_3 3+ 4 \log_3 3 \\ &= 5. \ _\square \end{align}\]

What is(are) the solution(s) of the quadratic equation \[\log 2x + \log(x-1) = \log\big(x^2+3\big) ?\] We have \[ \begin{align} \log 2x + \log(x-1) &= \log(x^2+3) \\ \log 2x(x-1) &= \log (x^2+3) \\ \Rightarrow 2x(x-1) &= x^2 +3 \\ x^2-2x-3 &= 0 \\ (x+1)(x-3) &= 0 \\ x &= -1, 3. \end{align} \] Since the logarithm functions \( \log(x-1)\) and \( \log 2x\) are defined over positive numbers, it must be true that \(x-1>0 \implies x>1\) and \(2x>0 \implies x>0.\) Thus, \(-1\) is can not be the value of \(x,\) implying that the value of \(x\) satisfying the given equation is \(x=3.\) \(_\square\)

What is the solution(s) of the quadratic equation \[ 2(\log x)^2 = 7\log x - 3 ?\] We have \[\begin{align} 2(\log x)^2 &= 7\log x - 3 \\ 2(\log x)^2 - 7\log x +3 &= 0 \\ (\log x -3)(2\log x -1) &= 0 \\ \Rightarrow \log x &= 3, \frac{1}{2} \\ x &= 1000, \sqrt{10}. \ _\square \end{align} \]

If the solutions of the quadratic equation \( x^{\log_3 x\,-\,2} = 27 \) are \(a\) and \(b,\) what is \( \log_{a} b + \log_{b} a?\) Taking logs with base 3 on both sides, we have \[ \begin{align} x^{\log_{3} x\,-\,2} &= 27 \\ \Rightarrow (\log_{3} x -2)\log_{3} x &= \log_{3} 27 \\ (\log_{3} x)^2-2\log_{3} x - 3 &= 0 \\ (\log_{3} x +1)(\log_{3} x - 3) &= 0 \\ \log_{3} x &= -1, 3. \end{align} \] Since \( \log_{a} b + \log_{b} a \) can be expressed as \(\frac{\log_{3} b}{\log_{3} a} + \frac{\log_{3} a}{\log_{3} b}\) using log with base 3, \[ \begin{align} \log_{a} b + \log_{b} a &= \frac{\log_{3} b}{\log_{3} a} + \frac{\log_{3} a}{\log_{3} b} \\ &= \frac{-1}{3} + \frac{3}{-1} \\ &= -\frac{10}{3}. \ _\square \end{align} \]

If the solutions of the equation \(\log_{2} x + a\log_{x} 8 = b \) are \(2\) and \(\frac{1}{8},\) what are \(a\) and \(b?\) We have \[ \begin{align} \log_{2} x + a\log_{x} 8 &= b \\ \log_{2} x + a\log_{x} 2^3 &= b \\ \log_{2} x + \frac{3a}{\log_{2} x} &= b \\ (\log_{2} x)^2 -b \log_{2} x + 3a &= 0. \qquad (1) \\ \end{align} \] Since the solutions of the equation \( {(\log_{2} x)}^2 -b \log_{2} x + 3a = 0 \) are \(2\) and \(\frac{1}{8} ,\) substituting \(2\) and \(\frac{1}{8}\) into \((1)\) gives \[ \begin{align} (\log_{2} 2)^2 - b \log_{2} 2 + 3a &= 0 \\ \Rightarrow 1-b+3a &= 0, \qquad (2)\\ (\log_{2} \frac{1}{8} )^2 - b \log_{2} \frac{1}{8} + 3a &= 0 \\ \Rightarrow 9+3b+3a &=0. \qquad (3) \end{align} \] Solving the simultaneous equations \((2)\) and \((3)\) gives \(a= -1\) and \(b = -2.\) \(_\square\)

The following logarithms are in an arithmetic progression:

\[\log_{2}4 + \log_{2}{16} + \log_{2}{64} + \cdots + x = 42.\]

If \(x\) can be expressed as \(\log_{2}a,\) find the value of \(a.\)

What are the solutions of the equation \[ \log_{x} xy \times \log_{y} xy + \log_{x}(x-y) \times \log_{y} (x-y) = 0?\] We have \[ \begin{align} \log_{x} xy \times \log_{y} xy + \log_{x}(x-y) \times \log_{y} (x-y) &= 0 \\ \frac{\log xy}{\log x} \times \frac{\log xy}{\log y} + \frac{\log(x-y)}{\log x} \times \frac{\log(x-y)}{\log y} &= 0 \\ \frac{(\log xy)^2 + (\log(x-y))^2}{\log x \cdot \log y} &= 0 \\ (\log xy)^2 + (\log(x-y))^2 &= 0 \\ \Rightarrow \log xy &= 0 \text{ and } \log(x-y)= 0. \\ \end{align} \] Since \(x\) and \(y\) are both positive, this implies that \[ \begin{align} xy &= 1 \text{ and } x-y=1 \\ \Rightarrow x&= \frac{\sqrt{5} +1}{2}, y=\frac{\sqrt{5}-1}{2}. \ _\square \end{align} \]

\[ \] Richter Scale:

Richter scale was developed by Charles Richter in 1935 to compare the intensities of earthquakes. The amount of energy released in an earthquake is very large, so a logarithmic scale avoids the use of large numbers.

The formula used for these calculations is

\[M= \log_{10}\left(\frac{I}{I_0}\right),\]

where \(M\) is the magnitude on the Richter scale, \(I\) is the intensity of the earthquake being measured, and \(I_0\) is the intensity of a reference earthquake.

Let's do a quick example to clarify how this works.

The 1906 San Francisco earthquake had a magnitude of 8.3 on the Richter scale. At the same time in South America there was an eathquake with magnitude 4.1 that caused only minor damage. How many times more intense was the San Francisco earthquake than the South American one?

Because the magnitude is a base-10 log, the Richter number is actually the exponent that 10 is raised to in order to calculate the intensity of the earthquake. Thus, the difference in magnitudes of the earthquakes can be calculated as follows:

\[M=\log_{10}\left(\frac{10^{8.3}}{10^{4.1}}\right)=4.2.\]

So, to answer the question, the San Francisco earthquake is more intense than the South American one by about \(10^M \approx 15848.93192\) times!

Note that you can just subtract 4.1 from 8.3 and get the same result. But if your math teachers are like mine, they will want you to use logarithms, and this is how it is done. The reason that subtracting the magnitudes works is because of the exponent rule for dividing exponents with the same base.

\[ \] Decibel Scale:

One decibel is one tenth of one bel, named in honor of Alexander Graham Bell. The bel is rarely used without the deci- prefix, deci- meaning one tenth. The decibel scale is used to calculate the difference in intensity between two sounds:

\[L=10\log_{10}\left(\frac{I}{I_0}\right),\]

where \( L\) is the loudness of the sound measured in decibels, \(I\) is the intensity of the sound being measured, and \(I_0\) is the intensity of the sound at the threshold of hearing which is equal to zero decibels.

\[ \] \(\text{pH}\) Scale:

The \(\text{pH}\) scale was invented in 1910 by Dr. Soren Sorenson, Head of Laboratory at Carlsberg Beer Company. The "H" in \(\text{pH}\) stands for hydrogen and the meaning of the "p" in \(\text{pH}\), although disputed, is generally considered to mean the power of hydrogen. This scale is used to measure the acidity or alkalinity of water or water soluble substances including, but definitely not limited to, soil or rainwater. The \(\text{pH}\) scale ranges from 1 to 14, where seven is a neutral point. Values below 7 indicate acidity with 1 being the most acidic. Values above 7 indicate alkalinity with14 being the most alkaline:

\[\text{pH}=-\log_{10}\ce{[H+]},\]

where \(\text{pH}\) is the \(\text{pH}\) number between \(1\) and \(14\) and \(\ce{[H+]}\) is the concentration of hydrogen ions.

• Solving Logarithmic Equations
• Graphs of Logarithmic Functions
• Solving Logarithmic Inequalities

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How to Solve Logarithms

Last Updated: March 17, 2024 Fact Checked

This article was reviewed by Grace Imson, MA . Grace Imson is a math teacher with over 40 years of teaching experience. Grace is currently a math instructor at the City College of San Francisco and was previously in the Math Department at Saint Louis University. She has taught math at the elementary, middle, high school, and college levels. She has an MA in Education, specializing in Administration and Supervision from Saint Louis University. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 260,631 times.

Logarithms might be intimidating, but solving a logarithm is much simpler once you realize that logarithms are just another way to write out exponential equations. Once you rewrite the logarithm into a more familiar form, you should be able to solve it as you would solve any standard exponential equation.

Before You Begin: Learn to Express a Logarithmic Equation Exponentially [1] X Research source [2] X Research source

• If and only if: b y = x
• b does not equal 1
• In the same equation, y is the exponent and x is the exponential expression that the logarithm is set equal to.

• Example: 1024 = ?

• This could also be written as: 4 5

• Example: 4 5 = 1024

Method One: Solve for X

• log 3 ( x + 5) + 6 - 6 = 10 - 6
• log 3 ( x + 5) = 4

• Comparing this equation to the definition [ y = log b (x) ], you can conclude that: y = 4; b = 3; x = x + 5
• Rewrite the equation so that: b y = x
• 3 4 = x + 5

• 3 * 3 * 3 * 3 = x + 5
• 81 - 5 = x + 5 - 5

• Example: x = 76

Method Two: Solve for X Using the Logarithmic Product Rule [3] X Research source [4] X Research source

• log b (m * n) = log b (m) + log b (n)

• log 4 (x + 6) + log 4 (x) = 2 - log 4 (x) + log 4 (x)
• log 4 (x + 6) + log 4 (x) = 2

• log 4 [(x + 6) * x] = 2
• log 4 (x 2 + 6x) = 2

• Comparing this equation to the definition [ y = log b (x) ], you can conclude that: y = 2; b = 4 ; x = x 2 + 6x
• 4 2 = x 2 + 6x

• 4 * 4 = x 2 + 6x
• 16 = x 2 + 6x
• 16 - 16 = x 2 + 6x - 16
• 0 = x 2 + 6x - 16
• 0 = (x - 2) * (x + 8)
• x = 2; x = -8

• Example: x = 2
• Note that you cannot have a negative solution for a logarithm, so you can discard x - 8 as a solution.

Method Three: Solve for X Using the Logarithmic Quotient Rule [5] X Research source

• log b (m / n) = log b (m) - log b (n)

• log 3 (x + 6) - log 3 (x - 2) = 2 + log 3 (x - 2) - log 3 (x - 2)
• log 3 (x + 6) - log 3 (x - 2) = 2

• log 3 [(x + 6) / (x - 2)] = 2

• Comparing this equation to the definition [ y = log b (x) ], you can conclude that: y = 2; b = 3; x = (x + 6) / (x - 2)
• 3 2 = (x + 6) / (x - 2)

• 3 * 3 = (x + 6) / (x - 2)
• 9 = (x + 6) / (x - 2)
• 9 * (x - 2) = [(x + 6) / (x - 2)] * (x - 2)
• 9x - 18 = x + 6
• 9x - x - 18 + 18 = x - x + 6 + 18
• 8x / 8 = 24 / 8

• Example: x = 3

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• ↑ https://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut43_logfun.htm#logdef
• ↑ https://www.mathsisfun.com/algebra/logarithms.html
• ↑ https://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut46_logeq.htm
• ↑ https://www.youtube.com/watch?v=fnhFneOz6n8
• ↑ https://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut44_logprop.htm

About This Article

To solve a logarithm, start by identifying the base, which is "b" in the equation, the exponent, which is "y," and the exponential expression, which is "x." Then, move the exponential expression to one side of the equation, and apply the exponent to the base by multiplying the base by itself the number of times indicated in the exponent. Finally, rewrite your final answer as an exponential expression. To learn how to solve for "x" in a logarithm, scroll down! Did this summary help you? Yes No

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• Solving Logarithmic Functions – Explanation & Examples

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Properties of logarithmic functions

Comparison of exponential function and logarithmic function, practice questions, solving logarithmic functions – explanation & examples.

Logarithms and exponents are two topics in mathematics that are closely related. Therefore it is useful we take a brief review of exponents.

An exponent is a form of writing the repeated multiplication of a number by itself. An exponential function is of the form f (x) = b y , where b > 0 < x and b ≠ 1. The quantity x is the number, b is the base, and y is the exponent or power.

For example , 32 = 2 × 2 × 2 × 2 × 2 = 2 2 .

On the other hand, the logarithmic function is defined as the inverse function of exponentiation. Consider again the exponential function f(x) = b y , where b > 0 < x and b ≠ 1. We can represent this function in logarithmic form as:

y = log b x

Then the logarithmic function is given by;

f(x) = log b x = y, where b is the base, y is the exponent, and x is the argument.

The function f (x) = log b x is read as “log base b of x.” Logarithms are useful in mathematics because they enable us to perform calculations with very large numbers.

How to Solve Logarithmic Functions?

To solve the logarithmic functions, it is important to use exponential functions in the given expression. The natural log or ln is the inverse of e . That means one can undo the other one i.e.

ln (e x ) = x

To solve an equation with logarithm(s), it is important to know their properties.

Properties of logarithmic functions are simply the rules for simplifying logarithms when the inputs are in the form of division, multiplication, or exponents of logarithmic values.

Some of the properties are listed below.

• Product rule

The product rule of logarithm states the logarithm of the product of two numbers having a common base is equal to the sum of individual logarithms.

⟹ log a  (p q) = log a  p + log a  q.

• Quotient rule

The quotient rule of logarithms states that the logarithm of the two numbers’ ratio with the same bases is equal to the difference of each logarithm.

⟹ log a  (p/q) = log a  p – log a q

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

⟹ log a  (p q ) = q log a p

• Change of Base rule

⟹ log a p = log x p ⋅ log a x

⟹ log q p = log x p / log x q

• Zero Exponent Rule

Other properties of logarithmic functions include:

• The bases of an exponential function and its equivalent logarithmic function are equal.
• The logarithms of a positive number to the base of the same number are equal to 1.

log a  a = 1

• Logarithms of 1 to any base are 0.

log a  1 = 0

• Log a 0 is undefined
• Logarithms of negative numbers are undefined.
• The base of logarithms can never be negative or 1.
• A logarithmic function with base 10is called a common logarithm. Always assume a base of 10 when solving with logarithmic functions without a small subscript for the base.

Whenever you see logarithms in the equation, you always think of how to undo the logarithm to solve the equation. For that, you use an exponential function . Both of these functions are interchangeable.

The following table tells the way of writing and interchanging the exponential functions and logarithmic functions . The third column tells about how to read both the logarithmic functions.

Let’s use these properties to solve a couple of problems involving logarithmic functions.

Rewrite exponential function 7 2 = 49 to its equivalent logarithmic function.

Given 7 2 = 64.

Here, the base = 7, exponent = 2 and the argument = 49. Therefore, 7 2 = 64 in logarithmic function is;

⟹ log 7 49 = 2

Write the logarithmic equivalent of 5 3 = 125.

exponent = 3;

and argument = 125

5 3 = 125 ⟹ log 5 125 =3

Solve for x in log  3  x = 2

log  3  x = 2 3 2  = x ⟹ x = 9

If 2 log x = 4 log 3, then find the value of ‘x’.

2 log x = 4 log 3

Divide each side by 2.

log x = (4 log 3) / 2

log x = 2 log 3

log x = log 3 2

log x = log 9

Find the logarithm of 1024 to the base 2.

1024 = 2 10

log 2 1024 = 10

Find the value of x in log 2 ( x ) = 4

Rewrite the logarithmic function log 2 ( x ) = 4 to exponential form.

Solve for x in the following logarithmic function log 2 (x – 1) = 5.

Solution Rewrite the logarithm in exponential form as;

log 2 (x – 1) = 5 ⟹ x – 1 = 2 5

Now, solve for x in the algebraic equation. ⟹ x – 1 = 32 x = 33

Find the value of x in log x 900 = 2.

Write the logarithm in exponential form as;

Find the square root of both sides of the equation to get;

x = -30 and 30

But since, the base of logarithms can never be negative or 1, therefore, the correct answer is 30.

Solve for x given, log x = log 2 + log 5

Using the product rule Log b  (m n) = log b  m + log b  n we get;

⟹ log 2 + log 5 = log (2 * 5) = Log   (10).

Therefore, x = 10.

Solve log  x  (4x – 3) = 2

Rewrite the logarithm in exponential form to get;

x 2  = 4x – 3

Now, solve the quadratic equation. x 2  = 4x – 3 x 2  – 4x + 3 = 0 (x -1) (x – 3) = 0

Since the base of a logarithm can never be 1, then the only solution is 3.

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Logarithmic Word Problems

Log Probs Expo Growth Expo Decay

What are logarithm word problems?

Logarithmic word problems, in my experience, generally involve either evaluating a given logarithmic equation at a given point, or else solving an equation for a given variable; they're pretty straightforward.

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What real-world problems use logarithms?

The classic real-world contexts for logarithm word problems are the measurement of acidity or alkalinity (that is, the measurement of pH), the measurement of sound (in decibels, or dB), and the measurement of earthquake intensity (on the Richter scale), among other uses ( link ).

Note: While log-based word problems are, in my experience, pretty straightforward, their statements tend to be fairly lengthy. Expect to have to plow through an unusual amount of text before they get to the point.

• Chemists define the acidity or alkalinity of a substance according to the formula pH =  −log[H + ] where [H + ] is the hydrogen ion concentration, measured in moles per liter. Solutions with a pH value of less than 7 are acidic; solutions with a pH value of greater than 7 are basic; solutions with a pH of 7 (such as pure water) are neutral.

a) Suppose that you test apple juice and find that the hydrogen ion concentration is [H + ] = 0.0003 . Find the pH value and determine whether the juice is basic or acidic.

b) You test some ammonia and determine the hydrogen ion concentration to be [H + ] = 1.3 × 10 −9 . Find the pH value and determine whether the ammonia is basic or acidic.

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In each case, I need to evaluate the pH function at the given value of [H + ] . In other words, this exercise, despite all the verbiage, is just plug-n-chug.

Since no base is specified, I will assume that the base for this logarithm is 10 , so that this is the so-called "common" log. (I happen to know that 10 is indeed the correct base, but they should have specified.)

a) In the case of the apple juice, the hydrogen ion concentration is [H + ] = 0.0003 , so:

pH = −log[H + ]

= −log[0.0003]

= 3.52287874528...

This value is less than 7 , so the apple juice is acidic.

b) In the case of the ammonia, the hydrogen ion concentration is [H + ] = 1.3 × 10 −9 , so:

= −log[1.3 × 10 −9 ] = 8.88605664769...

This value is more than 7 , so the ammonia is basic.

(a) The juice is acidic with a pH of about 3.5 , and (b) the ammonia is basic with a pH of about 8.9 .

When a logarithm is given without a base being specified, different people in different contexts will assume different bases; either 10 , 2 , or e . Ask now whether or not bases will be specified for all exercises, or if you're going to be expected to "just know" the bases for certain formulas, or if you're supposed to "just assume" that all logs without a specified base have a base of... [find out which one].

• "Loudness" is measured in decibels (abbreviated as dB). The formula for the loudness of a sound is given by dB = 10×log[I ÷ I 0 ] where I 0 is the intensity of "threshold sound", or sound that can barely be perceived. Other sounds are defined in terms of how many times more intense they are than threshold sound. For instance, a cat's purr is about 316 times as intense as threshold sound, for a decibel rating of:

dB = 10×log[I ÷ I 0 ]     = 10×log[ (316 I 0 ) ÷ I 0 ]     = 10×log[ 316 ]     = 24.9968708262...

...about 25 decibels.

Considering that prolonged exposure to sounds above 85 decibels can cause hearing damage or loss, and considering that a gunshot from a .22 rimfire rifle has an intensity of about I = (2.5 × 10 13 )I 0 , should you follow the rules and wear ear protection when practicing at the rifle range?

I need to evaluate the decibel equation at I = (2.5 × 10 13 )I 0 :

dB = 10log [ I ÷ I 0 ]     = 10log[ (2.5 ×10 13 )I 0 ÷ I 0 ]     = 10log[2.5 ×10 13 ]     = 133.979400087...

In other words, the squirrel gun creates a noise level of about 134 decibels. Since this is well above the level at which I can suffer hearing damage,

I should follow the rules and wear ear protection.

• Earthquake intensity is measured by the Richter scale. The formula for the Richter rating of a given quake is given by R = log[ I ÷ I 0  ] where I 0 is the "threshold quake", or movement that can barely be detected, and the intensity I is given in terms of multiples of that threshold intensity.

You have a seismograph set up at home, and see that there was an event while you were out that had an intensity of I = 989I 0 . Given that a heavy truck rumbling by can cause a microquake with a Richter rating of 3 or 3.5 , and "moderate" quakes have a Richter rating of 4 or more, what was likely the event that occurred while you were out?

To determine the probable event, I need to convert the intensity of the mystery quake into a Richter rating by evaluating the Richter function at I = 989I 0 :

R = log[ I ÷ I 0 ]     = log[ 989I 0 ÷ I 0 ]     = log[989]     = 2.9951962916...

A Richter rating of about 3 is not high enough to have been a moderate quake.

The event was probably just a big truck.

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'Being a good gravel racer is problem-solving' - Sarah Sturm on Unbound preparation, challenges in women's race

A year ago, Sarah Sturm (Specialized) finished third in the elite women’s category at Life Time Unbound Gravel 200, referring with high sarcasm her 12-plus hour adventure as “Funbound”. She said she was pleasantly surprised by her podium spot in the 200-mile, mud-soaked race, which actually came in at 205 miles, and “for me, that’s as good as winning”.

Two weeks prior to her third start at the giant of gravel gets underway on June 1, Sturm has already landed in Kansas and pre-ridden much of the north course, territory covered only twice before and last in 2021. She’s fresh off a podium at The Traka Gravel, the 360km distance, and ready to tackle the second round of the Life Time Grand Prix on another 11-plus hour adventure, where the unexpected is always to be expected - thunderstorms in the forecast, 2,000 more feet of elevation gain and a new test with a wider gap at the start between elite women and amateur racers.

“I'm kind of excited for the luck of the north course. I had a chance to go ride a lot of those [new] sections. And I don't know if it's helping with my gear selection or making more questions,” Sturm told a select group media, including Cyclingnews , in a pre-race virtual press conference this week. “But we have gone back and forth of what bike, what tires, to ride for the north. Pre-riding it is different than racing it.”

One of the talking points for Unbound Gravel 200 is the return to a northern journey across the undulating and unforgiving prairie of Kansas, approximately 92% of the 203-mile route comprised of rough roads strewn with tyre-slicing flint rocks plus elevation gain surpassing 11,000 feet. But what makes this even more challenging is the inevitable percolation with rain storms and winds across wide-open grasslands.

“I'm a fan of the mud and the elements and whatever Unbound has to bring. I like the problem-solving of gravel racing,” Sturm said, showing a wry smile as she explained why she loves the sport she adopted in 2019.

“I think some people think that it's because I'm trying to hide fitness or whatever, but I just think it's a huge part of being a good gravel racer is problem-solving and knowing how to fix a flat. You have to be really efficient at thinking through every single little detail. I've never raced pro road, but we can't put a hand up and ask for a bottle and ask for a new bike or a wheel.”

She has a diverse background in cycling, having competed as an amateur on the road, in mountain biking and cyclocross and a little on the track during her collegiate years at Fort Lewis College, where she intended to play soccer. A two-time US Single Speed Cyclocross National Champion (2018, 2019), she found herself standing on the podium at numerous gravel races in 2019, including a win at Belgian Waffle Ride California and third at SBT GRVL.

This season will be Sturm’s third campaign on the Life Time Grand Prix, her consistency rewarded with third overall in 2022 and fourth overall last year. She finished eighth in the opening event of the series, the 67-mile mountain bike race Sea Otter Classic Fuego XL. She mixed in a seventh at The Mid South gravel race and third at The Growler, a 140-mile, Classics-style road race in northern California.

Sturm said she expects the strongest field for any gravel race at this year’s Unbound, and she got an indication of the international talent headed to Kansas at The Traka 360 (228 miles) in Spain. After placing second in the women’s category with an effort just under 16 hours in the saddle last year, she returned in early May for the 360 again, which was modified the night before to 340km because of four days of heavy rain in advance of the race, and placed third. 

“This year in the 360, it was a much deeper, a more competitive field than the 200 [for the women] but unintentionally, it gave me sort of a sneak preview into what the competition for Unbound in the women's field is going to look like coming from Europe. I think the women's field is going to be one of the most competitive of Unbound that we've had. 

“The start of Traka started as quickly as Unbound will start. So I think that that's helpful. And knowing how riders do at the 11th hour, for me, that was helpful to see, like who has that extra kick.”

At The Traka, the distance and elevation gain of the long route was similar to what is on the menu for Unbound, moreso than any of the large, one-day gravel races on the calendar. Was it a good preparation for Kansas?

“The 360 at Traka was similar in elevation as Unbound will be, which is kind of wild to think about. The rolling hills at Unbound definitely stack up. There's a lot more singletrack and technical riding and diversity at Traka. I’m not a geologist, but there’s a lot of different types of rocks at Traka, and it seems there are fewer types at Unbound,” she said. 

“I definitely was surprised with the north course and its beauty and I just don't think that there's that same variation in type terrain, it's Kansas versus Gerona. The course does a good job of really testing the limits of everything.”

Separate race for women

Sturm said it is “a completely different landscape” in 2024 than when she switched to a gravel focus five years ago. While she was disappointed that Unbound did not implement drafting rules for the elite riders, related to riders from other categories, Life Time did modify the start of Unbound 200 to give the elite women a longer gap from the amateurs behind them on the start grid. 

Last year the amateurs started only 8 minutes after the elite women, and by the time the elites were slogging through the muddy climb at mile 11, the fields were mixed, so women-only tactics disappeared. 

“To be totally honest, it was a little disappointing to see that there was the no drafting policy that we thought was going to happen, not happen. It's a much more expensive, more logistically challenging discussion than just ‘OK, no drafting of men’. You know? I do think they’ll figure it out, hopefully, for next year,” Sturm said.

“At the end of the day, we just kind of want to see a deeper, more competitive women's field. And with that, I think team tactics could potentially be something in the future. I would love to be a part of that.”

On June 1, with the sun barely illuminating the throng of thousands gathered for multiple categories of races that begin on Commercial Street in downtown Emporia, the elite women will start at 6:05 a.m. CDT, five minutes after the elite men, and the amateur riders sett off 25 minutes later at 6:30 a.m. local time.

“The folks at Life Time and Unbound are helping us move closer and closer to get a fair race for women. It's a really tricky component of women's gravel racing, making it an extremely fair race for women. And right now we're not, we are just not quite there yet, but we're getting closer.

“But unfortunately, it'll probably be kind of going down to what group of dudes some of the women get into, just because that's how it always plays out, regardless of a separate start. But I mean, at the end of the day, that's the race that we have. So you play the game.”

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‘Ha Ha Ha Ha Ha Ha Ha’ Review: This Absurdist Clown Is Just Here to Help

A hit at Edinburgh Fringe last year, Julia Masli’s show arrives at SoHo Playhouse for its New York debut.

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By Elisabeth Vincentelli

For a show that has its audience in stitches, “Ha Ha Ha Ha Ha Ha Ha” is not without solemnity. On a recent evening, its sole performer, Julia Masli , called a spectator “the symbol of evil.” Another was “a symbol of the futility of mankind.” No matter: The crowd was doubled over from beginning to end. Was it the Estonia-born Masli’s strongly accented English? Her tone, which ranged from deceptively blank to deceptively sweet?

To be fair, these remarks landed in the general context of Masli trying to help people. In her breakthrough show , a hit last year at Edinburgh Fringe and now running at SoHo Playhouse in Manhattan, she goes up to audience members and simply asks, “Problem?” Then she proceeds to offer a solution.

Early in the evening, someone just as simply answered, “Sleep.” So Masli took him onstage, gave him an eye mask and had him lie down on a chaise longue, where he stayed for the remainder of the show. Another man revealed dating frustrations: “Gay men are insufferable,” he said. Masli appeared confounded, or at least acted that way, and replied, “I don’t know what to say.” Twice she made us hug our neighbors.

Moving up and down a single aisle with a discernible deliberateness, Masli projected a persona that was halfway between curious child and ingenuous alien just landed on Earth — that she is among us but not like us is reinforced by her having a golden mannequin leg for a left arm, with a mic attached at the end. Her otherworldliness is underlined by the work of the sound designer Alessio Festuccia and the sound tech Jonny Woolley, which creates an eerie mood that can turn discordant unexpectedly, and peaks in a fantastic coup de remix that shouldn’t be spoiled.

Masli wants to be of assistance, but her facade of naïveté leaves plenty of room for impishness. She is clown, comedian and trickster, revealing people to themselves and others, but also making them do her bidding. That last feat is quite impressive: The theatergoers may think of themselves as game for anything, but a more cynical observer might also marvel at the degree of obedience, and muse, “So that’s how cults are born.”

That dark undercurrent permeates the show, which Masli directed with the performance artist Kim Noble . It’s often poignant, sometimes grim, with stories of loneliness and estrangement repeatedly coming up at the performance I attended — people are asked about their problems, after all, not about their hobbies or achievements.

This theme and Masli’s Expressionist stage presence make “Ha Ha Ha Ha Ha Ha Ha” feel like comedy filtered through a European tradition of absurdist clowning that is not often experienced stateside, save for productions of “Waiting for Godot,” for example, or the family-friendly and sneakily melancholy “Slava’s Snowshow .” Yet Masli only dips a toe in those murky existential depths, before retreating into audience-friendly whimsy. Of course, it’s hard to complain that a show is too nice, but in this case it’s equally hard to shake the nagging feeling that “Ha Ha Ha Ha Ha Ha Ha” could be so much more.

Ha Ha Ha Ha Ha Ha Ha Through June 8 at SoHo Playhouse, Manhattan; sohoplayhouse.com . Running time: 1 hour 5 minutes.

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Packers Hope They’ve Solved Field-Tilting Watson’s Hamstring Problems

Bill huber | may 22, 2024.

• Green Bay Packers

GREEN BAY, Wis. – Green Bay Packers receiver Christian Watson’s hamstrings have been hamstrung by asymmetry.

The quest for answers to persistent hamstring issues led Watson, cornerback Eric Stokes and members of the team’s training staff to the University of Wisconsin, where they met with specialists from Badger Athletic Performance. Tests showed asymmetry in Watson’s hamstring. The left was significantly stronger and more powerful than the right hamstring.

“One, it puts strain on the left side, and the left is going through a lot more,” Watson explained at Packers OTAs on Tuesday. “And then, two, obviously, when you’re trying to be equal in power, it puts a lot more stress on the one that’s not as strong. So, that’s been the No. 1 thing for me because that leads to fatigue, as well. It’s a bad place to be so, obviously, that’s been my No. 1 goal to just kind of eliminate that.”

Watson said there was a “20-something percent” difference between the legs.

“Which is huge,” he said. It was a byproduct of injuries to his right hamstring that cost him three games in 2022, the first three games in 2023 and the final five games in 2023.

Through weekly treks to Madison and workouts in Green Bay, the difference has been cut in half.

“We have a NordBord in there that tells us the power outputs and the asymmetry,” Watson said. “Honestly, that’s my favorite part of the week, just going in there and getting to see that we’ve knocked off 4, 5, 6 percent of that asymmetry every week. Obviously, I still have a goal to get to. I want to be perfectly symmetrical, so I’ll continue to work that that.”

Perfect symmetry isn’t really possible – everyone has a dominant side. The goal is to find relative equilibrium. To get there, Watson does either an extra set of lifts or adds additional weight so his right leg is working harder than the left.

“You want to be within 6 percent symmetry on both legs,” Watson said. “Obviously, still working to get there.”

Watson was at his explosive best on Tuesday. He was arguably the best player on the practice field, making a number of catches 10, 15 and 20 yards down the field on passes from Jordan Love.

“I’m feeling really good,” he said.

Coach Matt LaFleur was cautiously optimistic that Watson has found the answers that will allow him to get on the field and stay on the field so he can stack day after day and then game after game.

“Time will tell. Certainly, you’ve got to get through the entire offseason, into training camp,” LaFleur said. “There’s a lot of volume in training camp, so we get through that, then we’ll feel pretty good about it. Both he and Stokes look like they’re probably in the best shape I’ve seen either one of them.”

The Packers have assembled a strong receiver corps. Last year, when Watson missed those eight games due to hamstring injuries, the Packers went 5-3. Moreover, passes thrown to Romeo Doubs, Jayden Reed, Dontayvion Wicks and Bo Melton resulted in passer ratings ranging from 109.5 to 124.5; passes to Watson resulted in a passer rating of 69.8. Of Love’s 12 interceptions on passes to receivers, six were directed to Watson.

Still, his importance to the team can’t be overstated because of his game-changing combination of speed and size.

“He’s incredibly bright, which allows us to move him all over the place,” LaFleur said. “Certainly, he brings the sheer size and physicality and speed that he possesses. You better know where he’s at at all times, because all it takes is one play. If he gets a sliver of light, he’s able to outrun everybody on the defense. So, yeah, he definitely changes and tilts the field in our favor when he’s out there.”

The key to tilting the field is staying on the field. As a rookie, he started his career by playing in two games and missing one, playing in two games and missing two. Through the team’s first nine games, Watson had 88 receiving yards. In Game 10 against Dallas and Game 12 against Philadelphia, Watson topped 100 yards. It was a similar story last year, with Watson turning in a series of ho-hum performances before catching five passes for 94 yards and one touchdown against Detroit and seven passes for 71 yards and two touchdowns against the Chiefs.

In retrospect, Watson said his issues were “pretty self-explanatory.” Now, the key will be eliminating the asymmetry and getting an even better understanding of his body to nip any potential problems in the bud. From that perspective, the trips to Madison have been enlightening.

“I liked science when I was a kid, but it was never my strong suit,” he said. “At this level of any sport, I think you’ve got to know your body. Just to be able to learn about it and understand how everything works has helped me a lot in terms of what I’m doing. I’m not just doing it because they’re telling me. I’m doing it because I understand what it’s going to do for me.”

More Green Bay Packers News

Two key members of the #Packers , Zach Tom and Tucker Kraft, will enter the 2024 NFL season with some lofty goals. For now, though, their focus will be on rehabbing their torn pectorals. ⬇️ https://t.co/Z78nryxPCi — Bill Huber (@BillHuberNFL) May 22, 2024

Packers OTAs: Big goals but torn pectorals | Jordan Love’s focus | Day 1 starters | NFL’s fastest | Five storylines | Incredible roster fact | Battles on defense | Battles on offense  

College coaches: Edgerrin Cooper | Javon Bullard | MarShawn Lloyd | Ty’Ron Hopper | Evan Williams | Jacob Monk | Kitan Oladapo | Michael Pratt | Kalen King

Bill Huber, who has covered the Green Bay Packers since 2008, is the publisher of Packer Central, a Sports Illustrated channel. E-mail: [email protected] History: Huber took over Packer Central in August 2019. Twitter: https://twitter.com/BillHuberNFL Background: Huber graduated from the University of Wisconsin-Whitewater, where he played on the football team, in 1995. He worked in newspapers in Reedsburg, Wisconsin Dells and Shawano before working at The Green Bay News-Chronicle and Green Bay Press-Gazette from 1998 through 2008. With The News-Chronicle, he won several awards for his commentaries and page design. In 2008, he took over as editor of Packer Report Magazine, which was founded by Hall of Fame linebacker Ray Nitschke, and PackerReport.com. In 2019, he took over the new Sports Illustrated site Packer Central, which he has grown into one of the largest sites in the Sports Illustrated Media Group.

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How to Talk to an Employee Who Isn’t Meeting Expectations

• Jenny Fernandez

It’s an opportunity to address the gap between the work they’re delivering and the company’s goals.

Approaching a conversation about improving an employee’s performance requires preparation, empathy, and a focus on collaboration. Even though hearing the truth about their current performance will be tough and potentially hurtful, it’s a teaching moment managers must embrace to help them become more resilient and adept at problem-solving and developing professional relationships. The author offers several strategies for treating difficult performance conversations not as fault-finding missions, but instead as opportunities to work collaboratively to define a shared commitment to growth and development.

As a leadership and team coach, I frequently encounter situations where managers feel ill-equipped to give their team members negative performance feedback. These conversations can be particularly challenging because the stakes are high for both sides. Unfavorable performance reviews and ratings come with tangible consequences for an employee’s compensation and career progression. Further, if the negative feedback is a surprise to them, it might prompt them to start looking for a new job.

• Jenny Fernandez , MBA, is an executive and team coach, Columbia and NYU faculty, and future of work and brand strategist. She works with senior leaders and their teams to become more collaborative, innovative, and resilient. Her work spans Fortune 500 companies, startups, and higher education. Jenny has been recognized by LinkedIn as a “Top Voice in Executive Coaching, Leadership Development, and Personal Branding” and was invited to join the prestigious Marshall Goldsmith’s 100 Coaches community. She is a Gen Z advocate. Connect with her on LinkedIn .

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Oxford geoscience professor myles allen on solving the problem of climate change.

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This week I had the pleasure of meeting Myles Allen . He is Head of Atmospheric, Oceanic and Planetary Physics in the Department of Physics, University of Oxford, and Professor of Geoscience in the School of Geography and Environment. He’s been studying how human and natural influences contribute to climate change since the early 1990s, served on the UN Intergovernmental Panel on Climate Change (IPCC) for its 3rd, 4th and 5th Assessments, and was a Coordinating Lead Author for its special report on 'the impacts of global warming of 1.5 °C above pre-industrial levels,’ and has been dubbed by the BBC “ the physicist behind net zero ”. In short, his scientific credentials regarding climate change are superb.

Professor Myles Allen Delivering The Gresham Lecture on May 21, 2024

But unlike many science professors, Professor Allen is bravely wading into the world of climate policy. The timing of my meeting with Professor Allen was fortuitous. It was the day before he gave a lecture on May 21, 2024, in his capacity as the Frank Jackson Professor of the Environment as part of the annual series of lectures hosted by Gresham College in the City of London. Gresham (b. 1519, d. 1579) was an English merchant, financier, and founder of the Royal Exchange. He started these lectures to ensure the emerging merchant classes were culturally enriched, as well as being versed in the latest science and technology.

The title of Professor Allen’s lecture is “ A Just and Inclusive Net Zero .” You can also view it online , which I highly recommend. The lecture is both global in its ideas and local in terms of the climate change debate taking place in the UK. I am less familiar with this than I am in the United States, and it was through this lens that I read and then listened to his lecture. My overall impression is how pragmatic and non-ideological it is. Professor Allen is no more an apologist for the fossil fuel industry than he is a supporter of the fossil fuel haters. Neither group brings value to the discussion of how to address climate change. Professor Allen does.

Sir Thomas Gresham

While I can’t do justice to his full lecture in this brief summary, here are four points that leaped out to me.

The first is he notes that “in the Paris-aligned scenarios of the IPCC, we are still using fossil fuels, at around one-quarter of the current rate, in 2100, long after the date of net zero.” Today’s world population of eight billion people will have grown to nearly 10 billion, with hopefully a larger percentage leading lives closer to what is enjoyed in the developed world. That means a LOT of fossil fuels. Those who harbor the fantasy that we still have time to achieve our climate goals simply by phasing out fossil fuel use need to get real. “One of the most dangerous myths is that achieving net zero is actually going to be really cheap because carbon-free, instantly-dispatchable energy” will soon take care of all of our energy needs. Nope. Even with the best of battery technologies it will be intermittent, and we’ll need natural gas and nuclear for the baseload.

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People also need to be realistic about some of the barriers to reducing fossil fuel use as rapidly as possible, with permitting being the number one in my mind. Environmentalists protesting transmission lines for bringing renewable power into the grid are part of the problem, not part of the solution. A great local example for me is the Northern Pass project for bringing cheap hydro power from Canada to Massachusetts, which I have written about with John Skjervem, the CIO of Utah Retirement Systems. Permitting extends to domestic mining of the cobalt and rare earths for battery and renewable energy technologies in order to ensure energy security and not being dependent on places like China, Russia, and the Democratic Republic of Congo for getting them.

Second, he calls out what he calls the “climate establishment” filled with those who regard themselves as elite experts who know better than the average citizen what needs to be done, such as “the unelected technocrats like the [UK] Climate Change Committee , the Science-Based Targets Initiative , or the Climate Action Tracker. ” Their perceived arrogance (whether it is real or not doesn’t matter) and disregard for which climate policies will be acceptable to those who will be subject to them is, again, part of the problem, not part of the solution. You can add to that bureaucrats in Brussels. As one example, he drolly quotes the Carbon Border Adjustment Mechanism, “or CBAM, is, in a nutshell, the European Union deciding that it has the right to impose punitive tariffs on imports from countries whose climate policies a team of bureaucrats in Brussels have decided aren’t good enough. I recently heard a talk from one such bureaucrat, I’m sure a very well-intentioned and intelligent chap, in which he said ‘of course, the CBAM is not neo-colonialist.’ If you have to assure people your policy is not neo-colonialist, you have a problem.”

Barnards Inn Hall, Gresham College, Where The Gresham Lecture Is Delivered

Professor Allen rightly suggests that these elite experts should “talking to people who wouldn’t normally show up in their social networks, like populist talk-show hosts, livestock farmers – and the executives of fossil fuel companies.” To that I’d like to add conservatives who are dedicating their lives to addressing climate change. On New Year’s Eve of 2023 I wrote about them and keep finding more. The “Eco-Right” has a very important role to play, and the climate establishment needs to show a little humility (yes, conservatives can have some very good ideas!) and be less self-righteous and start talking to them.

Third, while the “climate establishment” likes to talk about a Just Transition, it largely ignores what is just for the average citizen. Not everyone can go out and replace gas with heat pumps and buy an electric car. Yes, we need to think about a Just Transition for emerging markets, but in the U.S., we also need to think about what this means for low and middle income people in both red and blue states. Professor Allen also provocatively suggests “justice for the fossil fuel industry” which extends beyond “protecting workers in carbon intensive industries, or the interests of new fossil fuel producers, in ways that that just happens, surprise, surprise, to suit the fossil fuel industry itself rather well” to the shareholders of fossil fuel companies and people who benefit from their products—which is pretty much every single one of us.

Justice for the fossil fuel industry means acknowledging its right and need to exist but also holding it accountable for the carbon it produces. As Professor Allen notes, “The greatest climate injustice of all, to my mind, is the fact that the most profitable industry the world has ever known is entirely dependent on selling a product that is causing a very serious problem and no one is even asking them to fix it.” So how to fix it? Here are a few ways that spring to my mind of how not to fix it: (1) divesting from fossil fuel stocks in the naïve belief this will keep them from producing their product (but fine to do so if you don’t believe in the long-term value proposition or you’re just not ethically comfortable holding these stocks), (2) yelling at banks who provide them financing, (3) demanding that fossil fuel companies have unrealistic plans for reducing their investments and production, (4) urging them to get into the renewable energy business (their business model and capabilities don’t lend themselves to this), (5) opposing carbon capture storage technologies using the argument they just prolong the life of the fossil fuel industry, (6) thinking that reporting on Scope 3 emissions will somehow reduce the demand of their customers, and (7) filing useless shareholder proposals using the language of “value creation” to mask a basic hatred of the industry (which all the haters depend on).

Professor Myles Allen: “They just told me to write some physics on the board”

Which gets me to my fourth and last point. What is the solution? “There really is only one way to stop fossil fuels from causing global warming before the world stops using fossil fuels: we have to capture the carbon dioxide they generate and dispose of it, permanently, back underground.” The “underground” part is important. Turning fossil carbon into trees, water, and topsoil only delays its release into the atmosphere. The fossil fuel industry should take responsibility for doing this. It can start modestly, say one percent of the carbon it produces and building up to 100% by 2050. Let’s call this number the “geologically stored fraction.” This is what investors should be focused on, not Scope 3 emissions (although Scope 1 and 2 are fair game). Making this happen will require a mix of regulatory and market forces.

Can this be done? Absolutely. While fossil fuel companies don’t know wind and solar, they for sure know geological carbon management. The industry has a history of innovation and deep technological and engineering expertise for getting fossil fuels out of the ground, which often involving injecting stuff back into the ground. No question in my mind that it can figure out effective and cost efficient ways of putting carbon dioxide back underground at the scale required. In the beginning, the costs of doing so may be so small they just become a cost of doing business and preserving its needed license to operate. Ultimately, the costs will be borne by both the industry and its customers, which includes all of us. But if we want less carbon in the atmosphere but still want to enjoy the benefits of generating it, we must all be willing to pay our fair share for getting rid of it.

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Sam Altman's under pressure amid questions about OpenAI's commitment to safety

• OpenAI is facing a fresh wave of controversies.
• Former executives and AI experts have criticized the company's commitment to AI safety.
• As OpenAI races toward advanced AI, the pressure is starting to mount for Sam Altman.

AI's golden boy, Sam Altman, may be starting to lose his luster.

Under his stewardship, OpenAI is facing a fresh wave of controversy, including the use of tight nondisclosure agreements to silence employees and a public spat with a famous actor .

The company has also been dealing with comments from former executives that its commitment to AI safety leaves much to be desired.

Gretchen Krueger joined those ranks on Wednesday when the policy researcher announced she had parted ways with OpenAI.

Extinction risk

She said her decision to quit came before hearing that the senior executives Jan Leike and Ilya Sutskever had also resigned, but wrote on X that she had "overlapping concerns."

The public complaints from former workers are not a good look for a company at the forefront of developing technology that could have serious consequences for humanity.

Some of the leading experts in the field have long warned that advanced AI could pose an extinction risk to humanity — something Altman himself has noted.

Related stories

Stuart Russell, a leading AI researcher and a pioneer of the technology, told Business Insider that the race toward advanced AI could trigger anything from an explosion of AI deepfakes to an AI-led economic crash.

The University of California Berkeley professor labeled Altman's attitude toward building artificial general intelligence before figuring out how to make it safe as "completely unacceptable."

"This is why most of the safety people at OpenAI have left," Russell said, adding that tech companies across the board "were undermining every attempt at regulation."

"Even people who are developing the technology say there's a chance of human extinction. What gave them the right to play Russian roulette with everyone's children?"

ScarJo scandal

The criticism around AI safety is the latest blow for Altman, who is fighting battles on multiple fronts.

A recent tussle with the actor Scarlett Johansson over OpenAI's flagship voice for its new GPT-4o model has also hit a nerve with the creative community.

The actor lashed out at the company earlier this week, saying the voice sounded " eerily similar " to her own despite the fact she had turned down an offer from Sam Altman to work on the project.

Creatives have long accused AI companies of using their work without permission, and the Johansson debacle drove home the argument that OpenAI is running on an " ask forgiveness, not permission" model.

Amid all the drama, it's hard to overlook the controversy surrounding Altman since his dramatic ouster last November. He was suddenly removed as CEO by OpenAI's board, who cited his less than "candid" communication as part of the reason.

Altman ultimately won the high-stakes struggle — with a little help from Microsoft — and reclaimed the CEO role, as well as a new board and more power than ever.

However, recent reports suggest that under Altman's stewardship, OpenAI may not have been entirely truthful about how it handled NDAs for departing staff or transparent about how much of the company's resources are being directed toward AI safety initiatives. The company has been forced to change its policy on non-disparagement clauses for former employees.

As things continue to spiral for OpenAI, the outlook is looking a little bleak for Altman too.

Axel Springer, Business Insider's parent company, has a global deal to allow OpenAI to train its models on its media brands' reporting.

Watch: Sam Altman moves to Microsoft after OpenAI fires him as CEO

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Bobby Green: Paddy Pimblett not a bad fighter at all, but ‘the problem is he won’t shut the f*ck up’

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Bobby Green has watched his UFC 304 opponent Paddy Pimblett compete, and he saw a lot of things that were positive.

That is, until “The Baddy” opened his mouth.

“This is my next lesson for him: He’s solid, he’s not bad,” Green said on The MMA Hour ahead of UFC 304. “He’s not a bad fighter at all.

“The problem is, he won’t shut the f*ck up. Shut the f*ck up, smile, be that stupid kid with that stupid haircut, being happy and dancing, whatever he does. He does that little stupid dance, you know? Let him do his dance and be happy.”

Green and Pimblett meet at UFC’s July 27 pay-per-view in Manchester in a pivotal lightweight contest. Pimblett is undefeated inside the octagon, while Green makes the promotional walk for the 26th time, and does so coming off of a lopsided decision win over Jim Miller at UFC 300 .

In Green’s eyes, where Pimblett started to go wrong was after his controversial decision win over Jared Gordon at UFC 282 in December 2022.

“When [he started] talking, [then] that’s what got him all of this bullsh*t, Green said. “Shut the f*ck up, Paddy. Why? Because he started talking about the Jared [Gordon] fight. And not only that, he doubled down on that sh*t, like, ‘No, I whooped his ass.’

“Like, bro, everybody can see you lost that fight. And so, just be humble. He’s not humble, This sh*t’s going to humble him up.”

Since UFC 304’s announcement, UFC has faced a lot of local backlash due to the start time of the event, which caters more toward the U.S. audience than the Manchester crowd. At UFC 286 in March 2023 in London, the card took place on London time, and streamed in the afternoon in the U.S. This time around, the main card fighters are set to compete in the early morning hours, with the main event possibly starting around 5 a.m.

Green admits it’s not ideal, but he’ll be ready for Pimblett no matter what time it happens.

“We attack at dawn,” Green said. “I’ve been on my little Game of Thrones sh*t right now, and that’s where I’m on — like, we get the soldiers ready to wake up and go play and kill at dawn. In the middle of the dawn, 4 a.m., I’m getting up, and I’m going to have to kill him. That’s what we’re going to do.”

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