# 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 $b$ of $y$." 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 $\log_e{2}$.

Step-By-Step Example
Evaluating Logarithms
Evaluate $\log{1\text{,}000}$.
Step 1

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 $\log_b{y}=x$ to $b^x=y$.
\begin{aligned}\log_{10}{1\text{,}000}&=x\\10^x&=1\text{,}000\end{aligned}
Step 2
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.
Solution
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 $f(x)=b^x$.
Step-By-Step Example
Using the Inverse Function to Graph a Logarithmic Function
Graph the given logarithmic function and its inverse, and compare the graphs:
$f(x)=\log_2{x}$
Step 1

Write the equation of the inverse exponential function.

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

$f(x)=2^x$
Step 2

Complete a table of values for the inverse exponential function.

$x$
$f(x)=2^x$
$-2$
$0.25$
$-1$
$0.5$
$0$
$1$
$1$
$2$
$2$
$4$
Step 3

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

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

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 $f(x)=x$.