\chapter{Exponentials and logarithms}
\section{Exponential functions}
Up to this point we have studied the basic functions: powers, absolute value, square root,
cube root, polynomials, rational functions. With the exception of the absolute value function,
all these functions are obtained by combining together various power functions \(x^k\) with the
usual algebra operations.
So if we use only whole, non-negative numbers for the exponent we get the polynomial functions. If
we use the fractional values \(1/2\) and \(1/3\) we get the square root and cube root functions:
\[x^{1/2}=\sqrt{x}, \hspace{5ex} x^{1/3}=\sqrt[3]{x}.\]
If we use negative values we get rational functions:
\[x^{-1}=\frac{1}{x}, \hspace{5ex} x^{-2} = \frac{1}{x^2}.\]
In all these functions we always find the variable as the base, and a constant number in the
exponent.
In this chapter we will study different types of functions: those for which the variable is in
the exponent, and the base is a constant number. These functions are called \textit{exponential
functions}
\subsection{Irrational exponents}
But first we need to review the meaning of exponent. We know that \(3^4\) means \(3\times 3\times 3
\times 3\), \(3^{-1}\) means \(1/3\), and \(3^{1/2}\) means \(\sqrt{3}\). But in all these examples,
the exponent is a rational number. Since we are going to use a variable for the exponent, we need
to discuss what it means to have an exponent that is not a rational number, such as \(\sqrt{2}\).
\begin{example}
So what does \(3^{\sqrt{2}}\) really mean? To answer this question precisely requires Calculus.
In this Pre-Calculus course, we are going to use approximations, such as those we get with a
scientific calculator. Most calculators have a \(\fbox{\(y^x\)}\) key,
or a \(\fbox{ \(\wedge\) }\) key.
The \(\wedge\) symbol is for exponent.
In our examples we will show the steps using a TI 30X-IIS calculator. The steps may
be different for graphing calculators,
and also for some other models of scientific calculators.
To find \(3^{\sqrt{2}}\) we type
\[\fbox{3}\hspace{1ex}
\fbox{ \(\wedge\)} \hspace{1ex} \fbox{ \(\sqrt{^{}\hspace{0ex}} \)}(
\hspace{1ex} \fbox{ 2} \hspace{1ex} \fbox{ )} \hspace{1ex} \fbox{ =}\]
and we find the answer
\[3^{\sqrt{2}} =4.728804\ldots .\]
Note that we did not have to type the first open parenthesis, because the calculator automatically
includes it when you type the \(\fbox{\(\sqrt{^{}\hspace{0ex}}\)}\) key.
But we do not need to accept the calculator's answer without any questioning. After all, we know
that \(\sqrt{2}\) must be larger than \(1\) and smaller than \(2\) (because \(1^2=1\) is too small and \(2^2=4\) is too large):
\[1 < \sqrt{2} < 2,\]
and so we expect that \(3^{\sqrt{2}}\) is between \(3^1\) and \(3^2\):
\[3^1 < 3^{\sqrt{2}} < 3^2,\]
or
\[3 < 3^{\sqrt{2}} < 9,\]
in other words, \(3^{\sqrt{2}}\) is a number between \(3\) and \(9\),
and this agrees with what the calculator answer is.
\end{example}
\begin{example}
An irrational number we are familiar with is \(\pi = 3.1415\ldots\). Computing
\(2^\pi\) with a calculator we find
\[2^\pi =8.8249\ldots.\]
Since \(3 < \pi < 4\),
this agrees with
\[2^3 < 2^\pi < 2^4,\] that tells us that \(2^\pi\) is between \(8\) and \(16\).
\end{example}
\subsection{Using a scientific calculator}
We can use a scientific calculator to find approximations for more complicated expressions involving
irrational exponents. We give below the sequence of keystrokes needed for the
standard calculator used in this course.
Repeat the same sequence of steps with your calculator and check that you get the same answer.
\begin{example}
We want to find the approximate value of \(\displaystyle \frac{2^{\sqrt{5}}+1}{2^{\sqrt{5}}-1}\),
rounded to three decimal places.
This is the sequence of keystrokes we need:
\[\fbox{(} \hspace{0.7ex} \fbox{2} \hspace{0.7ex} \fbox{ \(\wedge\)} \hspace{0.7ex} \fbox{\(\sqrt{^{}\hspace{0ex}}\)}(
\hspace{0.7ex} \fbox{5} \hspace{0.7ex} \fbox{)} \hspace{0.7ex} \fbox{+} \hspace{0.7ex} \fbox{1}
\hspace{0.7ex} \fbox{)} \hspace{0.7ex} \fbox{\(\div\)} \hspace{0.7ex} \fbox{(}
\hspace{0.7ex} \fbox{2} \hspace{0.7ex} \fbox{ \(\wedge\)} \hspace{0.7ex} \fbox{\(\sqrt{^{}\hspace{0ex}}\)}(
\hspace{0.7ex} \fbox{5}
\hspace{0.7ex} \fbox{)} \hspace{0.7ex} \fbox{\(-\)} \hspace{0.7ex} \fbox{1} \hspace{0.7ex}
\fbox{)} \hspace{0.7ex} \fbox{=} \hspace{0.7ex} 1.539 .\]
Note the use of parentheses.
\end{example}
\subsection{Exponential functions}
We can now give our official definition of exponential function:
| Exponential function |
|---|
| An exponential function is one of the form |
| \(f(x) =b^x\) |
| where \(b\) is a positive number different from \(1\). |
We have already mentioned that the variable in the exponent can be any type of number:
positive, negative,
zero, rational, irrational. This means that the domain of an exponential function is
\(D=(-\infty, \infty)\).
The \(y\)-intercept is \((0,1)\), because \(f(0)=b^0=1\).
But it is impossible to solve \(b^x=0\).
If the exponent is negative, \(b^{-x}=1/b^x\) is still a positive number. So the function
has no \(x\)-intercepts.
Next, we investigate the shape of the graph. First we discuss
the case when the base is greater than \(1\).
\subsection{Graph of \(y=b^x\) when \(b> 1\)}
\begin{example}
Let \(f(x)=2^x\).
Using a calculator, we find several values of \(f(x)\):
\[
\begin{array}{c|ccccccccc}
x & -3 & -2 & -1 & -0.5 & 0 & 0.5 & 1 & 2 & 3 \\
\hline
2^x & 0.125 & 0.25 & 0.5 & 0.71 & 1 & 1.41 & 2 & 4 & 8
\end{array}
\]
The graph is shown in the picture below.
\[
\img{U3_1F1.png}{}{12em}{}
\]
Graph of \(y=2^x\)
We see from the graph that the function is always increasing, the range is \(R=(0,\infty)\),
and the \(x\)-axis is a horizontal
asymptote, with equation \(y=0\). The horizontal line test is clearly satisfied, so the
function is one-to-one and it has an inverse.
\end{example}
The shape and properties of the graph of any exponential function \(y=b^x\)
will be the same as those for \(y=2^x\), as long as the base \(b\) is greater than \(1\).
So for example the graph of \(y=3^x\) looks essentially the same, but it grows large faster
on the far right, and it goes to zero faster on the far left:
\[
\img{U3_1F2.png}{}{12em}{}
\]
Graphs of \(y=3^x\) and \(y=2^x\)
\subsection{The number \(e\)}
There is an important difference between the graphs of \(2^x\) and \(3^x\): if we draw a line with slope
\(1\) through the \(y\)-intercept (so the equation of the line will be \(y=x+1\)), the graph of \(2^x\) will cross it from above,
and the graph of \(3^x\) will cross it from below. This is hard to see unless
we zoom in and magnify the graph a great deal around the \(y\)-intercept, as shown in the following
picture:
\[ \img{U3_1F3.png}{}{15em}{}\]
dotted: \(y=2^x\) \(\hspace{2ex}\) dashed: \(y=3^x\) \(\hspace{2ex}\) solid: \(y=x+1\)
If we use a number slightly greater than \(2\) for the base, the graph will still cross the line
\(y=x+1\) from above, but at a slightly larger angle. And if we use a number slightly
less than \(3\), the graph will cross from below but at a slightly smaller angle.
There is a base between \(2\) and \(3\) for which the graph does not cross the line at all, but
just touches it, similar to a bouncing intercept of a polynomial functions. This number is
denoted by \(e\) and it is the most commonly used base for exponential functions. In fact,
it is common to refer to the function \(y=e^x\) as \textbf{the} exponential function. The number
\(e\) is irrational (like \(\pi\)), and its first few decimal digits are
\[e=2.718281828\ldots .\]
The graph of \(y=e^x\) is shown below, together with a magnification near the \(y\)-intercept:
\[
\img{U3_1F4.png}{}{12em}{} \hspace{1ex} \img{U3_1F5.png}{-1em}{12em}{}
\]
All scientific calculators have a key (usually labeled \(e^x\)) that allows us to find values of
\(e^x\). In our standard calculator, the \(\fbox{\(e^x\)}\) key will produce
\(e\hat{} (\) on the screen. The open parenthesis
is automatically included,
as in the \(\sqrt{^{}\hspace{0ex}}\) key.
\begin{example}
To find \(e^2\) we type
\[\fbox{\(e\hat{}\)}( \hspace{1ex}
\fbox{2}\hspace{1ex} \fbox{)} \hspace{1ex} \fbox{=} \]
and we find \(e^2=7.389056099\).
Using \(x=1\) allows us to find the number \(e\) itself, because \(e^1=e\):
\[\fbox{\(e\hat{}\)}( \hspace{1ex}
\fbox{1}\hspace{1ex} \fbox{)} \hspace{1ex} \fbox{=} \hspace{1ex} 2.718281828.\]
\end{example}
\begin{example}
We want to find the numerical value of
\(\displaystyle{\frac{3}{\sqrt{2e}}}\),
rounded to three decimal places.
\[ \mbox{To find } \frac{3}{\sqrt{2e}} \mbox{ type:} \hspace{5ex}
\fbox{3}\hspace{1ex} \fbox{\(\div\)} \hspace{1ex} \fbox{\(\sqrt{^{}\hspace{0ex}}\)}(\hspace{1ex}
\fbox{2}\hspace{1ex} \fbox{\(e^x \)}(\hspace{1ex} \fbox{1}\hspace{1ex} \fbox{)}\hspace{1ex}
\fbox{)} \hspace{1ex} \fbox{=} \hspace{1ex} 1.286645827 \]
\end{example}
We summarize below what we have found for the case \(b> 1\).
| Exponential function \(y=b^x\) with \(b> 1\)} |
|---|
|
\begin{itemize}
\item Examples: \(y=2^x\), \(y=3^x\), \(y=e^x\)
\item Domain \(=(-\infty,\infty)\), Range \(=(0,\infty)\).
\item \(y\)-intercept: \((0,1)\)
\item \(x\)-intercept: None
\item Horizontal asymptote: \(y=0\) (the \(x\)-axis)
\item Vertical asymptote: None
\item Always increasing
\item One-to-one: Yes
\item Shape of the graph:
\(\img{U3_1F6.png}{-5em}{12em}{}\)
\end{itemize}
|
\subsection{Graph of \(y=b^x\) when \(b< 1\)}
We now discuss the case that the base \(b\) is a positive number smaller than \(1\).
\begin{example}
We want to understand the graph of \(y=(1/2)^x\). Using the properties of exponents, we
find:
\[\left(\frac{1}{2}\right)^x =\frac{1}{2^x}=2^{-x}.\]
This means that if we start from the parent function \(f(x)=2^x\), then our function
is \(f(-x)\). So we can draw the graph just by reflecting \(2^x\) across the
\(y\)-axis:
\[
\begin{array}{ccc}
\img{U3_1F1.png}{}{12em}{}&
\longrightarrow &\img{U3_1F7.png}{}{12em}{}
\end{array}\]
We see that domain, range, intercepts and asymptotes remain the same, and the only difference
is that the graph is always decreasing.
It is common to write exponential functions whose
base is less than \(1\) by using the reciprocal of the base with a negative exponent.
So for example the number \(1/e\approx 0.367\) is less than \(1\), and instead of writing
\(y=\left(\frac{1}{e}\right)^x\) we will write \(y=e^{-x}\).
\end{example}
We summarize below the properties of \(y=b^x\) when \(b < 1\):
| Exponential function \(y=b^x\) with \(b< 1\) |
|---|
|
\begin{itemize}
\item Examples: \(y=2^{-x}\), \(y=3^{-x}\), \(y=e^{-x}\)
\item Domain \(=(-\infty,\infty)\), Range \(=(0,\infty)\).
\item \(y\)-intercept: \((0,1)\)
\item \(x\)-intercept: None
\item Horizontal asymptote: \(y=0\) (the \(x\)-axis)
\item Vertical asymptote: None
\item Always decreasing
\item One-to-one: Yes
\item Shape of the graph:
\(\img{U3_1F8.png}{-5em}{12em}{}\)
\end{itemize}
|
Problems
\problem
Find \(4^{\sqrt{5}}\) rounded to three decimal places. Then check that the answer is reasonable.
\begin{sol}
Typing
\[\fbox{4} \hspace{1ex} \fbox{ \(\wedge\) } \hspace{1ex} \fbox{\(\sqrt{^{\hspace{0ex}}} \)}(
\hspace{1ex} \fbox{5} \hspace{1ex} \fbox{)} \hspace{1ex}
\fbox{=}\]
we find
\[4^{\sqrt{5}} \approx 22.195.\]
We know that \(\sqrt{5}\) is about \(2.2\), and we can write
\[2 < \sqrt{5} < 3,\]
and so
\[4^2 < 4^{\sqrt{5}} < 4^3,\]
or
\[16 < 4^{\sqrt{5}} < 64.\]
So the answer given by the calculator is reasonable.
\end{sol}
\mproblem
Find \(3^{\sqrt{11}}\) rounded to three decimal places. Then check that the answer is reasonable.
\problem
Find the approximate value of the following numbers, rounded to three decimal places:
\begin{enumerate}
\item
\(\displaystyle \frac{2^{\pi/2}}{\pi-1}\)
\item
\(\displaystyle \frac{\pi^{\sqrt{3}}-1}{1+\sqrt{2}}\)
\end{enumerate}
\begin{sol}
\begin{enumerate}
\item \(\displaystyle \frac{2^{\pi/2}}{\pi-1}\)
\[ \fbox{2} \hspace{0.7ex} \fbox{ \(\wedge\)} \hspace{0.7ex} \fbox{(}
\hspace{0.7ex} \fbox{\(\pi\)} \hspace{0.7ex} \fbox{\(\div\)} \hspace{0.7ex} \fbox{2} \hspace{0.7ex} \fbox{)}
\hspace{0.7ex} \fbox{/} \hspace{0.7ex} \fbox{(}
\hspace{0.7ex} \fbox{\(\pi\)} \hspace{0.7ex} \fbox{\(-\)} \hspace{0.7ex} \fbox{1} \hspace{0.7ex} \fbox{)}
\hspace{0.7ex} \fbox{=}\hspace{0.7ex} 1.387 .\]
\item
\(\displaystyle \frac{\pi^{\sqrt{3}}-1}{1+\sqrt{2}}\)
\[ \fbox{(} \hspace{0.7ex} \fbox{\(\pi\)} \hspace{0.7ex} \fbox{ \(\wedge\)}
\hspace{0.7ex} \fbox{\(\sqrt{^{\hspace{0ex}}} \)}(
\hspace{0.7ex} \fbox{3} \hspace{0.7ex} \fbox{ ) } \hspace{0.7ex} \fbox{\(-\)}
\hspace{0.7ex} \fbox{1} \hspace{0.7ex} \fbox{)} \hspace{0.7ex} \fbox{\(\div\)}
\hspace{0.7ex} \fbox{(} \hspace{0.7ex} \fbox{1} \hspace{0.7ex} \fbox{\(+\)}
\hspace{0.7ex} \fbox{\(\sqrt{^{\hspace{0ex} }} \)}(\hspace{0.7ex} \fbox{2} \hspace{0.7ex}
\fbox{)} \hspace{0.7ex} \fbox{)} \hspace{0.7ex} \fbox{=} \hspace{0.7ex}2.594.\]
\end{enumerate}
\end{sol} \mproblem
Find the approximate value of the following numbers, rounded to three decimal places:
\begin{enumerate}
\item
\(\displaystyle \frac{3^{\sqrt{5}}}{\pi+2}\)
\item
\(\displaystyle \frac{2^{\sqrt{2}}+7}{\pi-\sqrt{2}}\)
\end{enumerate}
\problem
Find the numerical value of the expressions, rounded to three decimal places.
\begin{enumerate}
\item \(\displaystyle{\frac{1}{2\pi}+e^{-\frac{\pi}{2}}}\)
\item \(\displaystyle{\frac{e^{\sqrt{2\pi}}}{1-e^2}}\)
\end{enumerate}
\begin{sol}
\begin{enumerate}
\item
To find \(\dfrac{1}{2\pi}+e^{-\frac{\pi}{2}}\) type:
\[
\fbox{1}\hspace{0.7ex} \fbox{\(\div\)} \hspace{0.7ex} \fbox{(}\hspace{0.7ex}
\fbox{2}\hspace{0.7ex} \fbox{\(\pi\)}\hspace{0.7ex} \fbox{)}\hspace{0.7ex} \fbox{\(+\)}\hspace{0.7ex}
\fbox{\(e^x\)}( \hspace{0.7ex} \fbox{(-)} \hspace{0.7ex} \fbox{\(\pi\)} \hspace{0.7ex}
\fbox{\(\div\)} \hspace{0.7ex} \fbox{2} \hspace{0.7ex} \fbox{)} \hspace{0.7ex} \fbox{=} \hspace{0.7ex}
0.367034519.\]
\item
To find \( \dfrac{e^{\sqrt{2\pi}}}{1-e^2} \) type:
\[
\fbox{\(e^x\)}(\hspace{0.7ex} \fbox{\(\sqrt{^{\hspace{0ex}}}\)}( \hspace{0.7ex} \fbox{2}\hspace{0.7ex}
\fbox{\(\pi\)}\hspace{0.7ex} \fbox{)}\hspace{0.7ex} \fbox{)}\hspace{0.7ex} \fbox{\(\div\)}\hspace{0.7ex}
\fbox{(} \hspace{0.7ex} \fbox{1} \hspace{0.7ex} \fbox{\(-\)} \hspace{0.7ex}
\fbox{\(e^x\)}( \hspace{0.7ex} \fbox{2} \hspace{0.7ex} \fbox{)} \hspace{0.7ex} \fbox{)} \hspace{0.7ex}
\fbox{=} \hspace{0.7ex} -1.919455846.\]
\end{enumerate}
\end{sol} \mproblem
Find the numerical value of the expressions, rounded to three decimal places.
\begin{enumerate}
\item \(\displaystyle{e^{1-\sqrt{2}}-2^{e-1}}\)
\item \(\displaystyle{\frac{\pi^2}{2}+\frac{e}{\pi}}\)
\end{enumerate}