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Section 9.3 Logarithmic 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 FigureĀ 9.3.1 and several function values for \(f^{-1}\) are shown in FigureĀ 9.3.2

\(x\) \(y\)
\(-3\) \(0.125\)
\(-2\) \(0.25\)
\(-1\) \(0.5\)
\(0\) \(1\)
\(1\) \(2\)
\(2\) \(4\)
\(3\) \(8\)
\(4\) \(16\)
Figure 9.3.1. \(y=2^x\)
\(x\) \(y\)
\(0.125\) \(-3\)
\(0.25\) \(-2\)
\(0.5\) \(-1\)
\(1\) \(0\)
\(2\) \(1\)
\(4\) \(2\)
\(8\) \(3\)
\(16\) \(4\)
Figure 9.3.2. \(y=\log_2(x)\)

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

A graph of the functions \(y=2^x\) and \(y=\log_2(x)\text{.}\)  The dashed line \(y=x\) is also graphed.  The two functions are mirror images across the line \(y=x\text{.}\)  The function \(y=2^x\) is every increasing and concave up whereas the function \(y=\log_2(x)\) is everywhere concave down.  Every \(y\)-coordinate on the function \(y=2^x\) is positive whereas every \(x\)-coordinate on the function \(y=\log_2(x)\) is positive.  On the left side of the graph of \(y=2^x\) the \(x\)-axis acts as a horizontal asymptote whereas on the left side of the graph \(y=\log_2(x)\) the \(y\)-axis acts as a vertical asymptote.  As you look from left to right the function \(y=2^x\) starts out nearly horizontal and becomes more and more vertical whereas the function \(y=\log_2(x)\) starts out nearly vertical and becomes more and more horizontal.
Figure 9.3.3. \(\highlight{y=2^x}\,\,\text{and}\,\,\highlightr{y=\log_2(x)}\)

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 FigureĀ 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.

A graph of the functions \(y=0.5^x\) and \(y=\log_0.5(x)\text{.}\)  The dashed line \(y=x\) is also shown.  The two curves are mirror images across the line \(y=x\) and they intersect at the approximate point \((0.64,0.6)\text{.}\)  Both functions are everywhere decreasing and concave up.  The function \(y=0.5^x\) lies entirely above the \(x\)-axis while the function \(y=\log_0,5(x)\) lies entirely to the right of the \(y\)-axis.  The function \(y=0.5^x\) starts out nearly vertical and flattens out as it moves left-to-right ultimately approaching the positive \(x\)axis in an asymptotic fashion.  The function \(y=\log_05(x)\) start out nearly vertical to the right of the positive \(y\)-axis which acts as a vertical asymptote.  The function then becomes more and more horizontal as it moves rightward.
Figure 9.3.4. \(\highlight{y=0.5^x}\,\,\text{and}\,\,\highlightr{y=\log_{0.5}(x)}\)

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.