Relating Exponents to Logarithms

Logarithms and Exponents

Finding a logarithm means finding the exponent to which a base must be raised to equal a given value.

An exponent is a number that indicates how many times a base is multiplied by itself and shown by a raised number, such as:

4^3=4\cdot 4\cdot 4

In the exponential equation

y=b^x

b

is the base,

x

is the exponent, and

y

is the value of the power

b^x

. This equation can be read as "

y

equals

b

to the power of

x

." An example of an exponential equation in the same form is:

8 = 2^x

It can be read as "8 equals 2 to the power of

x

." Solving the equation involves finding the value of the exponent

x

to which the base 2 must be raised to equal 8. The equation, and others like it, can be solved by finding a logarithm. A logarithm is the exponent

x

to which a base

b

is raised to produce a given number

y

, written as:

x=\log_b{y}

The given number

y

is called the argument of the logarithm. The logarithmic equation can be read as "

x

equals the logarithm base

b

y

." It is equivalent to the exponential equation, where

b>0

and

b\neq 1

y=b^{x}

Exponential Form	Logarithmic Form
$y=b^x$	$x=\log_b{y}$
$8=2^x$	$x=\log_2{8}$
$8=2^3$	$3=\log_2{8}$

Some types of logarithms are used frequently. One is the common logarithm. A common logarithm is a logarithm with base 10, or the exponent to which 10 is raised to equal a given number; written as $\log{x}$ , where $x$ is the given number. Notice that a common logarithm can be written without a subscript indicating the base. For example, $\log{100}$ is the same as $\log_{10}{100}$ .

The irrational number $e$ is used frequently in logarithmic and exponential applications. Note that $e$ represents a constant and not a variable. The number $e$ is a nonterminating, nonrepeating decimal with a value of approximately 2.718281845.... A natural logarithm is a logarithm with base $e$ , or the exponent to which $e$ must be raised to equal a given number. A natural logarithm is written as $\ln{x}$ , where $x$ is the given number. So, $\ln{2}$ is the same as $\log_e{2}$ .

Evaluating Logarithms

Evaluate

\log{1\text{,}000}

Rewrite $\log{1\rm{,}000}$ in exponential form.

Since there is no base indicated, the expression is a common logarithm, meaning that the base is 10:

\log{1\text{,}000}=\log_{10}{1\text{,}000}

Let

x

represent the exponent, and convert the equation from the form

\log_b{y}=x

b^x=y

\begin{aligned}\log_{10}{1\text{,}000}&=x\\10^x&=1\text{,}000\end{aligned}

Calculate the value of

x

\begin{aligned}10^x&=1\text{,}000\\10^3&=1\text{,}000\end{aligned}

The value of

x

is 3.

Since the value of

x

is 3,

\log{1\text{,}000}

is equal to 3:

\log{1\text{,}000}=3

Technology is often required to evaluate logarithms. Since the base of a logarithm is restricted to positive real numbers other than 1, the number of possible logarithm bases is infinite. Common calculators come with two logarithmic functions: log for the common logarithm and ln for the natural logarithm. To evaluate

\log{900}

, enter log(900), which gives an approximate result of 2.95. To evaluate

\ln{900}

, enter ln(900), which gives an approximate answer of 6.80. Logarithms in other bases must be converted to either base 10 or base

e

to be evaluated with a calculator.

Inverses of Exponential Functions

A logarithmic function is the inverse of an exponential function.

An input is a value in the domain of a function. An output is a value in the range of a function. For an exponential function of the form $f(x)=b^x$ , the inputs of the function are exponents and the outputs are powers of the base $b$ .

An inverse function is the result of switching the inputs and outputs of a given function when the result is also a function. The composition of a function and its inverse is the identity function. If the inputs and outputs of the exponential $f(x)=b^x$ are switched to produce an inverse function, the inputs of the inverse function will be powers of the base $b$ , and the outputs will be exponents. In other words, the rule for the inverse function will be a logarithm.

A logarithmic function, where

b>0

and

b\neq 1

, has the form:

f(x)=\log_b{x}

A logarithmic function

f(x)=\log_b{x}

is the inverse of the exponential function

f(x)=b^x

The inverse function of an exponential function is its logarithmic function. For example, when 3 is the input for the exponential function

f(x)=2^x

, the output is 8. For the inverse logarithmic function

f(x)=\log_2{x}

, the relationship is reversed. When 8 is the input, the output is 3.

Using the Inverse Function to Graph a Logarithmic Function

Graph the given logarithmic function and its inverse, and compare the graphs:

f(x)=\log_2{x}

Write the equation of the inverse exponential function.

The inverse of the given logarithmic function is the exponential function:

f(x)=2^x

Complete a table of values for the inverse exponential function.

$x$	$f(x)=2^x$
$-2$	$0.25$
$-1$	$0.5$
$0$	$1$
$1$	$2$
$2$	$4$

Complete a table of values for the logarithmic function by switching the input and output values for the inverse exponential function.

$x$	$f(x)=\log_2{x}$
$0.25$	$-2$
$0.5$	$-1$
$1$	$0$
$2$	$1$
$4$	$2$

Plot ordered pairs from the table for the inverse exponential function:

f(x)=2^x

Then, connect them with a smooth curve.

Plot ordered pairs from the table for the logarithmic function:

f(x)=\log_2{x}

Then, connect them with a smooth curve.

The curves are reflections of each other across the diagonal line $f(x)=x$ .

<Vocabulary >Properties of Logarithms

Logarithmic Functions

Contents

Relating Exponents to Logarithms

Logarithms and Exponents

Inverses of Exponential Functions