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Section 3.1 Logarithms as Inverses of Exponential Functions

Technology advances at exponential rates, and human institutions and societies do not. They adapt at much slower rates. Those gaps get wider and wider. — Mitchell “Mitch” Kapor, A pioneer of the personal computing industry and advocate for social change, 1950 –

Problem: Solve

\begin{equation*} 10^x=3. \end{equation*}

Reminder:For all \(a,b\in (0,1)\cup (1,\infty )\) and all \(x,y\in \mathbb{R}\text{:}\)

  1. \(\displaystyle a^{x+y}=a^x\cdot a^y\)

  2. \(\displaystyle a^{x-y}=\ds \frac{a^x}{a^y}\)

  3. \(\displaystyle \left( a^x\right) ^y=a^{xy}\)

  4. \(\displaystyle (ab)^x=a^x\cdot b^x\)

Getting there: Let

\begin{equation*} f(x)=2^x \text{ and } g(x)=\left(\frac{1}{2}\right)^x. \end{equation*}

Complete the following table:

\begin{equation*} \begin{array}{c||r|r| r|r|r|r|r|} x\amp-3\amp-2\amp-1\amp0\amp1\amp2\amp3\\ \hline \hline f(x)\amp \amp \amp \amp \amp \amp \amp\\ \hline g(x)\amp \amp \amp \amp \amp \amp \amp\\ \end{array} \end{equation*}
Solution

Observe:

\begin{equation*} \begin{array}{c||r|r| r|r|r|r|r|} x\amp -3\amp -2\amp -1\amp0\amp1\amp2\amp3\\ \hline \hline f(x)\amp 2^{-3}=\frac{1}{8} \amp 2^{-2}=\frac{1}{4} \amp 2^{-1}=\frac{1}{2}\amp2^{0}=1\amp 2^{1}=2\amp 2^{2}=4\amp2^{3}=8\\ \hline g(x)\amp\left(\frac{1}{2}\right)^{-3}=8 \amp \left(\frac{1}{2}\right)^{-2}=4 \amp \left(\frac{1}{2}\right)^{-1}=2\amp \left(\frac{1}{2}\right)^{0}=1 \amp \left(\frac{1}{2}\right)^{1}=\frac{1}{2} \amp \left(\frac{1}{2}\right)^{2}=\frac{1}{4}\amp\left(\frac{1}{2}\right)^{3}=\frac{1}{8}\\ \end{array} \end{equation*}

Two exponential functions: See Figure 3.1.

Figure 3.1. Two exponential functions: \(\displaystyle f(x)=2^x\) and \(\displaystyle g(x)=\left( \frac{1}{2}\right)^x\text{.}\)

Graphs of exponential functions - general case: See Figure 3.2.

Figure 3.2. Graphs of exponential functions: \(0\lt a\lt 1\) vs. \(a\gt 1\text{.}\)

Summary:

\begin{equation*} \begin{array}{l|l|l} f(x)=a^x\amp 0\lt a\lt 1\amp a\gt 1\\ \hline \text{Domain}\amp \mathbb{R}\amp \mathbb{R}\\ \hline \text{Range} \amp (0,\infty)\amp (0,\infty)\\ \hline f(x)=0 \amp \text{Never}\amp \text{Never}\\ \hline f(x)=1 \amp x=0\amp x=0\\ \hline \text{Increasing}\amp -- \amp \mathbb{R}\\ \hline \text{Decreasing}\amp \mathbb{R}\amp --\\ \hline \text{One-to-one}\amp \text{Yes}\amp \text{Yes}\\ \hline \text{Odd or even}\amp \text{Neither}\amp \text{Neither}\\ \hline \text{Horizontal asymptote}\amp y=0\amp y=0\\ \end{array} \end{equation*}

Big questions: What is the inverse function of \(\displaystyle f(x)=a^x\text{,}\) \(a\gt 0\text{,}\) \(a\not= 1\text{?}\)

See Figure 3.3.

Figure 3.3. Graph of the inverse function of an exponential function.

Logarithmic Function. The inverse function of the exponential function \(f(x)=a^x\text{,}\) \(a\gt 0\text{,}\) \(a\not=1\text{,}\) is called the logarithmic function with base \(a\) and it is denoted by

\begin{equation*} f^{-1}(x)=\log _a(x). \end{equation*}

All You Need To Know. For any \(a\gt 0\text{,}\) \(a\not= 1\text{,}\) any \(x\gt 0\text{,}\) and any \(y\in \mathbb{R}\)

\begin{equation*} \log _a(x)=y \ \Leftrightarrow \ a^y=x. \end{equation*}
Example 3.1.1. Determine logarithm.

  1. Determine \(\log_2{(16)}$, $\log_2{(\frac{1}{8})}$ and $\log_2{(1)}\text{.}\)

  2. Can you find \(\log_2{(-32)}\text{?}\)

  3. Solve \(\displaystyle10^x=3\)

Solution

  1. By definition

    \begin{equation*} \log_2{(16)}=t\Leftrightarrow 16=2^t \Leftrightarrow t=4\Leftrightarrow \log_2{(16)}=4, \end{equation*}
    \begin{equation*} \log_2{(\frac{1}{8})}=u\Leftrightarrow \frac{1}{8}=2^u \Leftrightarrow 2^{-3}=2^u\Leftrightarrow u=-3\Leftrightarrow \log_2{(\frac{1}{8})}=-3, \end{equation*}
    \begin{equation*} \log_2{(1)}=v\Leftrightarrow 1=2^v \Leftrightarrow t=0\Leftrightarrow \log_2{(1)}=0. \end{equation*}

  2. Note that the range of the function \(f(x)=2^x\) is the set of all positive real numbers. This implies that the domain of the function \(f^{-1}(x)=\log_2(x)\) is the set of all positive real numbers. Thus \(\log_2{(-32)}\) is not defined.

  3. By definition \(x=\log_{10}(3)\text{.}\)

Note: If \(a=10\) then

\begin{equation*} \log_{10}(x)=\log (x). \end{equation*}

The logarithm base \(10\) is called the common logarithm.

Logarithmic functions. See Figure 3.4.

Figure 3.4. Graphs of logarithmic functions: \(0\lt a\lt 1\) vs. \(a\gt 1\text{.}\)

Summary:

\begin{equation*} \begin{array}{l|l|l} f(x)=\log_a x\amp 0\lt a\lt 1\amp a\gt 1\\ \hline \text{Domain}\amp (0,\infty)\amp (0,\infty)\\ \hline \text{Range} \amp \mathbb{R}\amp \mathbb{R}\\ \hline f(x)=0 \amp x=1\amp x=-\\ \hline f(x)=1 \amp x=a\amp x=a\\ \hline \text{Increasing}\amp -- \amp(0,\infty)\\ \hline \text{Decreasing}\amp (0,\infty)\amp --\\ \hline \text{One-to-one}\amp \text{Yes}\amp \text{Yes}\\ \hline f^{-1}(x)=\amp a^x\amp a^x\\ \hline \text{Odd or even}\amp \text{Neither}\amp \text{Neither}\\ \hline \text{Vertical Asymptote}\amp x=0\amp x=0\\ \end{array} \end{equation*}