## Section9.3Logarithmic Functions and Their Graphs

The inverse of a simple exponential function, $f(x)=b^x, b \gt 0, b \neq 1\text{,}$ is called a logarithmic function. The inverse of $f(x)=b^x$ is symbolized as

\begin{equation*} f^{-1}(x)=\log_b(x) \end{equation*}

which is read aloud as "f-inverse of x equals the log-base-b of x."

Let's consider the function $f(x)=2^x$ and its inverse $f^{-1}(x)=\log_2(x)\text{.}$ Recall that if $(a,b)$ is an ordered pair on $f\text{,}$ then $(b,a)$ is an ordered pair on $f^{-1}\text{.}$ Several function values for $f$ are shown in Figure 9.3.1 and several function values for $f^{-1}$ are shown in Figure 9.3.2

Let's observe that every ordered pair in Figure 9.3.2 satisfies the $2^y=x\text{.}$ It follows that

\begin{equation*} y=\log_2(x)\,\,\text{if and only if}\,\,2^y=x. \end{equation*}

This relationship is explored in depth in the section about logarithms as exponents. In the current context, the relationship is a direct result of the inverse function relationship between exponential and logarithmic functions. For the function $f(x)=2^x\text{,}$ the ordered pairs satisfy the equation $y=2^x\text{.}$ It immediately follows that the ordered pairs on $f^{-1}$ satisfy the equation $2^y=x\text{.}$

Plots of both $f(x)=2^x$ and $f^{-1}(x)=\log_2(x)$ are plotted in Figure 9.3.3 . The line $y=x$ is also plotted to highlight the fact that the two functions are reflections of one another across that line. Let's first notice that the range of $f(x)=2^x\text{,}$ $(0,\infty)\text{,}$ becomes the domain of $f^{-1}(x)=\log_2(x)\text{.}$ Similarly, the domain of $f(x)=2^x\text{,}$ $(-\infty,\infty)\text{,}$ becomes the range of $f^{-1}(x)=\log_2(x)\text{.}$

Additional properties of $f(x)=2^x$ and their attendant properties of $f^{-1}(x)=\log_2(x)$ are shown below. Please note that these properties are true whenever the base, $b\text{,}$ is greater than 1. Note that the symbols $x \rightarrow 0^+$ are read as "x approaches zero from the right."

Properties of $f(x)=2^x$

• The domain is $(-\infty,\infty)\text{.}$

• The range is $(0,\infty)\text{.}$

• $f(0)=1$
• As $x \rightarrow -\infty\text{,}$ $y \rightarrow 0$

• As $x \rightarrow \infty\text{,}$ $y \rightarrow \infty$

Properties of $f^{-1}(x)=\log_2(x)$

• The range is $(-\infty,\infty)\text{.}$

• The domain is $(0,\infty)\text{.}$

• $f^{-1}(1)=0$
• As $x \rightarrow 0^+\text{,}$ $y \rightarrow -\infty$

• As $x \rightarrow \infty\text{,}$ $y \rightarrow \infty$

Plots of both $f(x)=0.5^x$ and $f^{-1}(x)=\log_{0.5}(x)$ are plotted in Figure 9.3.4 . These shapes are representative of $y=b^x$ and $y=\log_b(x)$ whenever $0 \lt b \lt 1\text{.}$ The properties are summarized below.

Properties of $f(x)=0.5^x$

• The domain is $(-\infty,\infty)\text{.}$

• The range is $(0,\infty)\text{.}$

• $f(0)=1$
• As $x \rightarrow -\infty\text{,}$ $y \rightarrow \infty$

• As $x \rightarrow \infty\text{,}$ $y \rightarrow 0$

Properties of $f^{-1}(x)=\log_{0.5}(x)$

• The range is $(-\infty,\infty)\text{.}$

• The domain is $(0,\infty)\text{.}$

• $f^{-1}(1)=0$
• As $x \rightarrow \infty\text{,}$ $y \rightarrow -\infty$

• As $x \rightarrow 0^+\text{,}$ $y \rightarrow \infty$

One closing comment. In the section about exponential functions and their graphs, it was discussed that on a graph of $f(x)=2^x\text{,}$ with a scale of inches on each axis, before $x$ even gets to three feet, the value of $y$ has already passed the moon. That means that on a similarly scaled graph of $f^{-1}(x)=\log_2(x)\text{,}$ $x$ will have to extend beyond the moon before the corresponding value of $y$ will even get three feet off the ground! So while the range of $y=\log_2(x)$ is $(-\infty,\infty)\text{,}$ the spectacularly slow pace at which the function increases as $x \rightarrow \infty$ makes the upper "limit" of the range somewhat misleading.