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Section7.3Logarithmic 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 Table 7.3.1 and several function values for \(f^{-1}\) are shown in Table 7.3.2

\(x\) \(y\)
\(-3\) \(0.125\)
\(-2\) \(0.25\)
\(-1\) \(0.5\)
\(0\) \(1\)
\(1\) \(2\)
\(2\) \(4\)
\(3\) \(8\)
\(4\) \(16\)
\(x\) \(y\)
\(0.125\) \(-3\)
\(0.25\) \(-2\)
\(0.5\) \(-1\)
\(1\) \(0\)
\(2\) \(1\)
\(4\) \(2\)
\(8\) \(3\)
\(16\) \(4\)

Let's observe that every ordered pair in Table 7.3.2 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 7.3.3 . 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 \(f^{-1}(x)=\log_2(x)\text{.}\)

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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 7.3.4 . 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.

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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.