Logarithmic Functions

Learning Objectives

Convert between exponential and logarithmic form. (IA 10.3.1)
Evaluate logarithmic functions. (IA 10.3.2)

Objective 1: Convert between exponential and logarithmic form. (IA 10.3.1)

Practice Makes Perfect

Graph the exponential function

f (x) = 2^{x}

by making a table.

.
$x$	$y = f (x)$

A blank Cartesian coordinate system with labeled x and y axes ranging from -10 to 10, complete with grid lines.

Is it one-to-one?
Domain?
Range?
Graph the inverse of $f (x) = 2^{x}$ on the grid above by interchanging x and y coordinates in the table.
.

$x$ $y = f (x)$
Is the inverse one-to-one function?
Domain?
Range?

Example 1

Find the inverse of $f (x) = 2^{x}$

Solution

.
Rewrite with $y = f (x)$	$y = 2^{x}$
Interchange the variables $x$ and $y$ .	$x = 2^{y}$
Solve for $y$ .	Oops! We have no way to solve for $y$ .

We give y a new notation:

y = \log_{x} 2

y = \log_{x} 2

" read "the logarithm, base 2 of x", means "the power to which we raise 2 to get x". The function

y = \log_{a} x

is equivalent to

a^{y} = x

is the logarithmic function with base

a

, where

a > 0

x > 0

Since the equations $y = {log}_{a} x$ and $x = a^{y}$ are equivalent, we can go back and forth between them. This will often be the method to solve some exponential and logarithmic equations. To help with converting back and forth, let’s take a close look at the equations. Notice the positions of the exponent and base.

This figure shows the expression y equals log sub a of x, where y is the exponent and a is the base. Next to this expression we have x equals a to the y, where again y is the exponent and a is the base. — Figure 1

If we remember the logarithm is the exponent, it makes the conversion easier. You may want to repeat, “base to the exponent gives us the number.”

Example 2

Convert between exponential and logarithmic form.

ⓐ Convert to logarithmic form: $2^{3} = 8$

Solution

Identify the base and the exponent: the base is 2 and the exponent is 3.
Then we have $3 = {log}_{2} 8$ .

ⓑ Convert to exponential form: ${log}_{b} m = a$

Solution

Identify the base and the exponent: the base is b and the exponent is a.
Then we have $b^{a} = m$ .

Practice Makes Perfect

Convert between exponential and logarithmic form.

Remember these logarithmic notations to help complete the following:
Common Logarithm $log x = {log}_{x} 10$
Natural Logarithm $ln x = {log}_{x} e$

Convert to logarithmic form.

ⓐ $8 = 2^{x}$
ⓑ $10^{-2} = 0.01$
ⓒ $e^{x} = 40$

Convert to exponential form.

ⓐ $\log_{81} 3 = 4$
ⓑ $\ln 1 = 0$
ⓒ $\log 10000 = 4$

Objective 2: Evaluate logarithmic functions (IA 10.3.2).

We can solve and evaluate logarithmic equations by using the technique of converting the equation to its equivalent exponential form.

Example 3

Find the value of x: ⓐ ${log}_{x} 36 = 2,$ ⓑ ${log}_{4} x = 3,$ and ⓒ ${log}_{\frac{1}{2}} \frac{1}{8} = x .$

Solution

ⓐ

.
	${log}_{x} 36 = 2$
Convert to exponential form.	$x^{2} = 36$
Solve the quadratic.	$x = 6, x = −6$
The base of a logarithmic function must be positive, so we eliminate $x = −6$ .	$x = 6 Therefore, {log}_{6} 36 = 2 .$

ⓑ

.
	${log}_{4} x = 3$
Convert to exponential form.	$4^{3} = x$
Simplify.	$x = 64 Therefore, {log}_{4} 64 = 3 .$

ⓒ

.
	${log}_{\frac{1}{2}} \frac{1}{8} = x$
Convert to exponential form.	${(\frac{1}{2})}^{x} = \frac{1}{8}$
Rewrite $\frac{1}{8}$ as ${(\frac{1}{2})}^{3}$ .	${(\frac{1}{2})}^{x} = {(\frac{1}{2})}^{3}$
With the same base, the exponents must be equal.	$x = 3 Therefore, {log}_{\frac{1}{2}} \frac{1}{8} = 3$

Practice Makes Perfect

Evaluate logarithmic functions.

Find the value of

x

ⓐ $\log_{25} x = 2$
ⓑ $\log_{x} 4 = 2$
ⓒ $\log_{x} \frac{1}{3} = 2$

Evaluate each of the following.

ⓐ $\log_{100} 10$
ⓑ $\log 0.1$
ⓒ $\log_{2} 4$
ⓓ $\log_{1} 20$
ⓔ $\log_{4} 4$
ⓕ $\log_{\frac{1}{9}} 3$
ⓖ $\log_{\sqrt{2}} 2$
ⓗ $\ln e^{- 5}$

Photo of the aftermath of the earthquake in Japan with a focus on the Japanese flag. — Figure 2 Devastation of March 11, 2011 earthquake in Honshu, Japan. (credit: Daniel Pierce)

In 2010, a major earthquake struck Haiti, destroying or damaging over 285,000 homeshttp://earthquake.usgs.gov/earthquakes/eqinthenews/2010/us2010rja6/#summary. Accessed 3/4/2013.. One year later, another, stronger earthquake devastated Honshu, Japan, destroying or damaging over 332,000 buildings,http://earthquake.usgs.gov/earthquakes/eqinthenews/2011/usc0001xgp/#summary. Accessed 3/4/2013. like those shown in Figure 2. Even though both caused substantial damage, the earthquake in 2011 was 100 times stronger than the earthquake in Haiti. How do we know? The magnitudes of earthquakes are measured on a scale known as the Richter Scale. The Haitian earthquake registered a 7.0 on the Richter Scalehttp://earthquake.usgs.gov/earthquakes/eqinthenews/2010/us2010rja6/. Accessed 3/4/2013. whereas the Japanese earthquake registered a 9.0.http://earthquake.usgs.gov/earthquakes/eqinthenews/2011/usc0001xgp/#details. Accessed 3/4/2013.

The Richter Scale is a base-ten logarithmic scale. In other words, an earthquake of magnitude 8 is not twice as great as an earthquake of magnitude 4. It is $10^{8 - 4} = 10^{4} = 10,000$ times as great! In this lesson, we will investigate the nature of the Richter Scale and the base-ten function upon which it depends.

Converting from Logarithmic to Exponential Form

In order to analyze the magnitude of earthquakes or compare the magnitudes of two different earthquakes, we need to be able to convert between logarithmic and exponential form. For example, suppose the amount of energy released from one earthquake were 500 times greater than the amount of energy released from another. We want to calculate the difference in magnitude. The equation that represents this problem is $10^{x} = 500,$ where $x$ represents the difference in magnitudes on the Richter Scale. How would we solve for $x ?$

We have not yet learned a method for solving exponential equations. None of the algebraic tools discussed so far is sufficient to solve $10^{x} = 500.$ We know that $10^{2} = 100$ and $10^{3} = 1000,$ so it is clear that $x$ must be some value between 2 and 3, since $y = 10^{x}$ is increasing. We can examine a graph, as in Figure 3, to better estimate the solution.

Graph of the intersections of the equations y=10^x and y=500. — Figure 3

Estimating from a graph, however, is imprecise. To find an algebraic solution, we must introduce a new function. Observe that the graph in Figure 3 passes the horizontal line test. The exponential function $y = b^{x}$ is one-to-one, so its inverse, $x = b^{y}$ is also a function. As is the case with all inverse functions, we simply interchange $x$ and $y$ and solve for $y$ to find the inverse function. To represent $y$ as a function of $x,$ we use a logarithmic function of the form $y = \log_{b} (x) .$ The base $b$ logarithm of a number is the exponent by which we must raise $b$ to get that number.

We read a logarithmic expression as, “The logarithm with base $b$ of $x$ is equal to $y,$ ” or, simplified, “log base $b$ of $x$ is $y .$ ” We can also say, “ $b$ raised to the power of $y$ is $x,$ ” because logs are exponents. For example, the base 2 logarithm of 32 is 5, because 5 is the exponent we must apply to 2 to get 32. Since $2^{5} = 32,$ we can write $\log_{2} 32 = 5.$ We read this as “log base 2 of 32 is 5.”

We can express the relationship between logarithmic form and its corresponding exponential form as follows:

\log_{b} (x) = y \Leftrightarrow b^{y} = x, b > 0, b \neq 1

Note that the base $b$ is always positive.

Visualizing the conversion from logarithmic form log_b(x) = y to exponential form b^y = x, with a helpful arrow diagram and the phrase "Think b to the y = x".

Because logarithm is a function, it is most correctly written as $\log_{b} (x),$ using parentheses to denote function evaluation, just as we would with $f (x) .$ However, when the input is a single variable or number, it is common to see the parentheses dropped and the expression written without parentheses, as $\log_{b} x .$ Note that many calculators require parentheses around the $x .$

We can illustrate the notation of logarithms as follows:

This image shows the fundamental relationship between logarithms and exponents, stating that log_b(c) = a is equivalent to b^a = c, visually guided by orange circular arrows.

Notice that, comparing the logarithm function and the exponential function, the input and the output are switched. This means $y = \log_{b} (x)$ and $y = b^{x}$ are inverse functions.

Example 4

Converting from Logarithmic Form to Exponential Form

Write the following logarithmic equations in exponential form.

ⓐ $\log_{6} (\sqrt{6}) = \frac{1}{2}$
ⓑ $\log_{3} (9) = 2$

Solution

First, identify the values of $b, y, and x .$ Then, write the equation in the form $b^{y} = x .$

ⓐ $\log_{6} (\sqrt{6}) = \frac{1}{2}$
Here, $b = 6, y = \frac{1}{2}, and x = \sqrt{6.}$ Therefore, the equation $\log_{6} (\sqrt{6}) = \frac{1}{2}$ is equivalent to $6^{\frac{1}{2}} = \sqrt{6} .$
ⓑ $\log_{3} (9) = 2$
Here, $b = 3, y = 2, and x = 9.$ Therefore, the equation $\log_{3} (9) = 2$ is equivalent to $3^{2} = 9.$

Converting from Exponential to Logarithmic Form

To convert from exponents to logarithms, we follow the same steps in reverse. We identify the base $b,$ exponent $x,$ and output $y .$ Then we write $x = \log_{b} (y) .$

Example 5

Converting from Exponential Form to Logarithmic Form

Write the following exponential equations in logarithmic form.

$2^{3} = 8$
$5^{2} = 25$
$10^{- 4} = \frac{1}{10,000}$

Solution

First, identify the values of $b, y, and x .$ Then, write the equation in the form $x = \log_{b} (y) .$

$2^{3} = 8$
Here, $b = 2,$ $x = 3,$ and $y = 8.$ Therefore, the equation $2^{3} = 8$ is equivalent to $\log_{2} (8) = 3.$
$5^{2} = 25$
Here, $b = 5,$ $x = 2,$ and $y = 25.$ Therefore, the equation $5^{2} = 25$ is equivalent to $\log_{5} (25) = 2.$
$10^{- 4} = \frac{1}{10,000}$
Here, $b = 10,$ $x = - 4,$ and $y = \frac{1}{10,000} .$ Therefore, the equation $10^{- 4} = \frac{1}{10,000}$ is equivalent to ${log}_{10} (\frac{1}{10,000}) = - 4.$

Evaluating Logarithms

Knowing the squares, cubes, and roots of numbers allows us to evaluate many logarithms mentally. For example, consider $\log_{2} 8.$ We ask, “To what exponent must $2$ be raised in order to get 8?” Because we already know $2^{3} = 8,$ it follows that $\log_{2} 8 = 3.$

Now consider solving $\log_{7} 49$ and $\log_{3} 27$ mentally.

We ask, “To what exponent must 7 be raised in order to get 49?” We know $7^{2} = 49.$ Therefore, $\log_{7} 49 = 2$
We ask, “To what exponent must 3 be raised in order to get 27?” We know $3^{3} = 27.$ Therefore, $\log_{3} 27 = 3$

Even some seemingly more complicated logarithms can be evaluated without a calculator. For example, let’s evaluate $\log_{\frac{2}{3}} \frac{4}{9}$ mentally.

We ask, “To what exponent must $\frac{2}{3}$ be raised in order to get $\frac{4}{9} ?$ ” We know $2^{2} = 4$ and $3^{2} = 9,$ so ${(\frac{2}{3})}^{2} = \frac{4}{9} .$ Therefore, $\log_{\frac{2}{3}} (\frac{4}{9}) = 2.$

Example 6

Solving Logarithms Mentally

Solve $y = \log_{4} (64)$ without using a calculator.

Solution

First we rewrite the logarithm in exponential form: $4^{y} = 64.$ Next, we ask, “To what exponent must 4 be raised in order to get 64?”

We know

4^{3} = 64

Therefore,

\log {}_{4}{(64)} = 3

Example 7

Evaluating the Logarithm of a Reciprocal

Evaluate $y = \log_{3} (\frac{1}{27})$ without using a calculator.

Solution

First we rewrite the logarithm in exponential form: $3^{y} = \frac{1}{27} .$ Next, we ask, “To what exponent must 3 be raised in order to get $\frac{1}{27} ?$ ”

We know $3^{3} = 27,$ but what must we do to get the reciprocal, $\frac{1}{27} ?$ Recall from working with exponents that $b^{- a} = \frac{1}{b^{a}} .$ We use this information to write

\begin{array}{l} 3^{- 3} = \frac{1}{3^{3}} \\ = \frac{1}{27} \end{array}

Therefore, $\log_{3} (\frac{1}{27}) = - 3.$

Using Common Logarithms

Sometimes you may see a logarithm written without a base. When you see one written this way, you need to look at the expression before evaluating it. It may be that the base you use doesn't matter. If you find it in computer science, it often means $\log_{2} (x)$ . However, in mathematics it almost always means the common logarithm of 10. In other words, the expression $\log (x)$ often means $\log_{10} (x) .$

Currently, we use $\log_{b} (x), \lg (x)$ as the common logarithm, $lb (x)$ as the binary logarithm, and $\ln (x)$ as the natural logarithm. Writing $\lg (x)$ without specifying a base is now considered bad form, despite being frequently found in older materials.

Example 8

Finding the Value of a Common Logarithm Mentally

Evaluate $y = \log (1000)$ without using a calculator.

Solution

First we rewrite the logarithm in exponential form: $10^{y} = 1000.$ Next, we ask, “To what exponent must $10$ be raised in order to get 1000?” We know

10^{3} = 1000

Therefore, $\log (1000) = 3.$

Example 9

Finding the Value of a Common Logarithm Using a Calculator

Evaluate $y = \log (321)$ to four decimal places using a calculator.

Solution

Press [LOG].
Enter 321, followed by [ ) ].
Press [ENTER].

Rounding to four decimal places, $\log (321) \approx 2.5065.$

Analysis

Note that $10^{2} = 100$ and that $10^{3} = 1000.$ Since 321 is between 100 and 1000, we know that $\log (321)$ must be between $\log (100)$ and $\log (1000) .$ This gives us the following:

\begin{matrix} 100 & < & 321 & < & 1000 \\ 2 & < & 2.5065 & < & 3 \end{matrix}

Example 10

Rewriting and Solving a Real-World Exponential Model

The amount of energy released from one earthquake was 500 times greater than the amount of energy released from another. The equation $10^{x} = 500$ represents this situation, where $x$ is the difference in magnitudes on the Richter Scale. To the nearest thousandth, what was the difference in magnitudes?

Solution

We begin by rewriting the exponential equation in logarithmic form.

\begin{array}{l} 10^{x} & = 500 \\ \log (500) & = x & Use the definition of the common log . \end{array}

Next we evaluate the logarithm using a calculator:

Press [LOG].
Enter $500,$ followed by [ ) ].
Press [ENTER].
To the nearest thousandth, $\log (500) \approx 2.699.$

The difference in magnitudes was about $2.699.$

Using Natural Logarithms

The most frequently used base for logarithms is $e,$ the value of which is approximately $2.71828$ . Base $e$ logarithms are important in calculus and some scientific applications; they are called natural logarithms. The base $e$ logarithm, $\log_{e} (x),$ has its own notation, $\ln (x) .$

Most values of $\ln (x)$ can be found only using a calculator. The major exception is that, because the logarithm of 1 is always 0 in any base, $\ln 1 = 0.$ For other natural logarithms, we can use the $\ln$ key that can be found on most scientific calculators. We can also find the natural logarithm of any power of $e$ using the inverse property of logarithms.

Example 11

Evaluating a Natural Logarithm Using a Calculator

Evaluate $y = \ln (500)$ to four decimal places using a calculator.

Solution

Press [LN].
Enter $500,$ followed by [ ) ].
Press [ENTER].

Rounding to four decimal places, $\ln (500) \approx 6.2146$

Key Equations

...
Definition of the logarithmic function	For $x > 0, b > 0, b \neq 1,$ $y = \log_{b} (x)$ if and only if $b^{y} = x .$
Definition of the common logarithm	For $x > 0,$ $y = \log (x)$ if and only if $10^{y} = x .$
Definition of the natural logarithm	For $x > 0,$ $y = \ln (x)$ if and only if $e^{y} = x .$

Key Concepts

The inverse of an exponential function is a logarithmic function, and the inverse of a logarithmic function is an exponential function.
Logarithmic equations can be written in an equivalent exponential form, using the definition of a logarithm. See Example 4.
Exponential equations can be written in their equivalent logarithmic form using the definition of a logarithm See Example 5.
Logarithmic functions with base $b$ can be evaluated mentally using previous knowledge of powers of $b .$ See Example 6 and Example 7.
Common logarithms can be evaluated mentally using previous knowledge of powers of $10.$ See Example 8.
When common logarithms cannot be evaluated mentally, a calculator can be used. See Example 9.
Real-world exponential problems with base $10$ can be rewritten as a common logarithm and then evaluated using a calculator. See Example 10.
Natural logarithms can be evaluated using a calculator Example 11.

Section Exercises

Verbal

Exercise 1

What is a base $b$ logarithm? Discuss the meaning by interpreting each part of the equivalent equations $b^{y} = x$ and $\log_{b} x = y$ for $b > 0, b \neq 1.$

Solution

A logarithm is an exponent. Specifically, it is the exponent to which a base $b$ is raised to produce a given value. In the expressions given, the base $b$ has the same value. The exponent, $y,$ in the expression $b^{y}$ can also be written as the logarithm, $\log_{b} x,$ and the value of $x$ is the result of raising $b$ to the power of $y .$

Exercise 2

How is the logarithmic function $f (x) = \log_{b} x$ related to the exponential function $g (x) = b^{x} ?$ What is the result of composing these two functions?

Exercise 3

How can the logarithmic equation $\log_{b} x = y$ be solved for $x$ using the properties of exponents?

Solution

Since the equation of a logarithm is equivalent to an exponential equation, the logarithm can be converted to the exponential equation $b^{y} = x,$ and then properties of exponents can be applied to solve for $x .$

Exercise 4

Discuss the meaning of the common logarithm. What is its relationship to a logarithm with base $b,$ and how does the notation differ?

Exercise 5

Discuss the meaning of the natural logarithm. What is its relationship to a logarithm with base $b,$ and how does the notation differ?

Solution

The natural logarithm is a special case of the logarithm with base $b$ in that the natural log always has base $e .$ Rather than notating the natural logarithm as $\log_{e} (x),$ the notation used is $\ln (x) .$

Algebraic

For the following exercises, rewrite each equation in exponential form.

Exercise 6

${log}_{4} (q) = m$

Exercise 7

${log}_{a} (b) = c$

Solution

$a^{c} = b$

Exercise 8

$\log_{16} (y) = x$

Exercise 9

$\log_{x} (64) = y$

Solution

$x^{y} = 64$

Exercise 10

$\log_{y} (x) = −11$

Exercise 11

$\log_{15} (a) = b$

Solution

$15^{b} = a$

Exercise 12

$\log_{y} (137) = x$

Exercise 13

$\log_{13} (142) = a$

Solution

$13^{a} = 142$

Exercise 14

$log (v) = t$

Exercise 15

$ln (w) = n$

Solution

$e^{n} = w$

For the following exercises, rewrite each equation in logarithmic form.

Exercise 16

$4^{x} = y$

Exercise 17

$c^{d} = k$

Solution

${log}_{c} (k) = d$

Exercise 18

$m^{- 7} = n$

Exercise 19

$19^{x} = y$

Solution

$\log_{19} y = x$

Exercise 20

$x^{- \frac{10}{13}} = y$

Exercise 21

$n^{4} = 103$

Solution

$\log_{n} (103) = 4$

Exercise 22

${(\frac{7}{5})}^{m} = n$

Exercise 23

$y^{x} = \frac{39}{100}$

Solution

$\log_{y} (\frac{39}{100}) = x$

Exercise 24

$10^{a} = b$

Exercise 25

$e^{k} = h$

Solution

$ln (h) = k$

For the following exercises, solve for $x$ by converting the logarithmic equation to exponential form.

Exercise 26

${log}_{3} (x) = 2$

Exercise 27

${log}_{2} (x) = - 3$

Solution

$x = 2^{- 3} = \frac{1}{8}$

Exercise 28

${log}_{5} (x) = 2$

Exercise 29

$\log_{3} (x) = 3$

Solution

$x = 3^{3} = 27$

Exercise 30

${log}_{2} (x) = 6$

Exercise 31

${log}_{9} (x) = \frac{1}{2}$

Solution

$x = 9^{\frac{1}{2}} = 3$

Exercise 32

${log}_{18} (x) = 2$

Exercise 33

$\log_{6} (x) = - 3$

Solution

$x = 6^{- 3} = \frac{1}{216}$

Exercise 34

$log (x) = 3$

Exercise 35

$ln (x) = 2$

Solution

$x = e^{2}$

For the following exercises, use the definition of common and natural logarithms to simplify.

Exercise 36

$log (100^{8})$

Exercise 37

$10^{log (32)}$

Solution

$32$

Exercise 38

$2 log (.0001)$

Exercise 39

$e^{\ln (1.06)}$

Solution

$1.06$

Exercise 40

$\ln (e^{- 5.03})$

Exercise 41

$e^{\ln (10.125)} + 4$

Solution

$14.125$

Numeric

For the following exercises, evaluate the base $b$ logarithmic expression without using a calculator.

Exercise 42

${log}_{3} (\frac{1}{27})$

Exercise 43

${log}_{6} (\sqrt{6})$

Solution

$\frac{1}{2}$

Exercise 44

${log}_{2} (\frac{1}{8}) + 4$

Exercise 45

$6 {log}_{8} (4)$

Solution

$4$

For the following exercises, evaluate the common logarithmic expression without using a calculator.

Exercise 46

$log (10, 000)$

Exercise 47

$log (0.001)$

Solution

$- 3$

Exercise 48

$log (1) + 7$

Exercise 49

$2 log (100^{- 3})$

Solution

$- 12$

For the following exercises, evaluate the natural logarithmic expression without using a calculator.

Exercise 50

$ln (e^{\frac{1}{3}})$

Exercise 51

$ln (1)$

Solution

$0$

Exercise 52

$ln (e^{- 0.225}) - 3$

Exercise 53

$25 ln (e^{\frac{2}{5}})$

Solution

$10$

Technology

For the following exercises, evaluate each expression using a calculator. Round to the nearest thousandth.

Exercise 54

$log (0.04)$

Exercise 55

$ln (15)$

Solution

$2.708$

Exercise 56

$ln (\frac{4}{5})$

Exercise 57

$log (\sqrt{2})$

Solution

$0.151$

Exercise 58

$ln (\sqrt{2})$

Extensions

Exercise 59

Is $x = 0$ in the domain of the function $f (x) = \log (x) ?$ If so, what is the value of the function when $x = 0 ?$ Verify the result.

Solution

No, the function has no defined value for $x = 0.$ To verify, suppose $x = 0$ is in the domain of the function $f (x) = \log (x) .$ Then there is some number $n$ such that $n = \log (0) .$ Rewriting as an exponential equation gives: $10^{n} = 0,$ which is impossible since no such real number $n$ exists. Therefore, $x = 0$ is not the domain of the function $f (x) = \log (x) .$

Exercise 60

Is $f (x) = 0$ in the range of the function $f (x) = \log (x) ?$ If so, for what value of $x ?$ Verify the result.

Exercise 61

Is there a number $x$ such that $\ln x = 2 ?$ If so, what is that number? Verify the result.

Solution

Yes. Suppose there exists a real number $x$ such that $\ln x = 2.$ Rewriting as an exponential equation gives $x = e^{2},$ which is a real number. To verify, let $x = e^{2} .$ Then, by definition, $\ln (x) = \ln (e^{2}) = 2.$

Exercise 62

Is the following true: $\frac{\log_{3} (27)}{\log_{4} (\frac{1}{64})} = −1 ?$ Verify the result.

Exercise 63

Is the following true: $\frac{\ln (e^{1.725})}{\ln (1)} = 1.725 ?$ Verify the result.

Solution

No; $\ln (1) = 0,$ so $\frac{\ln (e^{1.725})}{\ln (1)}$ is undefined.

Real-World Applications

Exercise 64

The exposure index $E I$ for a camera is a measurement of the amount of light that hits the image receptor. It is determined by the equation $E I = \log_{2} (\frac{f^{2}}{t}),$ where $f$ is the “f-stop” setting on the camera, and $t$ is the exposure time in seconds. Suppose the f-stop setting is $8$ and the desired exposure time is $2$ seconds. What will the resulting exposure index be?

Exercise 65

Refer to the previous exercise. Suppose the light meter on a camera indicates an $E I$ of $- 2,$ and the desired exposure time is 16 seconds. What should the f-stop setting be?

Solution

$2$

Exercise 66

The intensity levels I of two earthquakes measured on a seismograph can be compared by the formula $\log \frac{I_{1}}{I_{2}} = M_{1} - M_{2}$ where $M$ is the magnitude given by the Richter Scale. In August 2009, an earthquake of magnitude 6.1 hit Honshu, Japan. In March 2011, that same region experienced yet another, more devastating earthquake, this time with a magnitude of 9.0.http://earthquake.usgs.gov/earthquakes/world/historical.php. Accessed 3/4/2014. How many times greater was the intensity of the 2011 earthquake? Round to the nearest whole number.

common logarithm

the exponent to which 10 must be raised to get

x;

\log_{10} (x)

is written simply as

\log (x) .

logarithm

the exponent to which

b

must be raised to get

x;

written

y = \log_{b} (x)

natural logarithm

the exponent to which the number

e

must be raised to get

x;

\log_{e} (x)

is written as

\ln (x) .