Precalculus Mathematics for Calculus

James Stewart, Lothar Redlin, Saleem Watson

Chapter 4

Exponential and Logarithmic Functions - all with Video Answers

Educators

Section 1

Exponential Functions

Problem 1

The function $f(x)=5^{x}$ is an exponential function with base _____; $f(-2)=$ _____, $f(0)=$ _____, $f(2)=$ _____, and $f(6)=$ _____.

Jeffrey Russell

Jeffrey Russell

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Problem 2

Match the exponential function with one of the graphs labeled I, II, III, or IV, shown below.
$$\begin{array}{ll}{\text { (a) } f(x)=2^{x}} & {\text { (b) } f(x)=2^{-x}} \\ {\text { (c) } f(x)=-2^{x}} & {\text { (d) } f(x)=-2^{-x}}\end{array}$$

Jeffrey Russell

Jeffrey Russell

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Problem 3

(a) To obtain the graph of $g(x)=2^{x}-1,$ we start with the graph of $f(x)=2^{x}$ and shift it _____ (upward/downward) 1 unit.
(b) To obtain the graph of $h(x)=2^{x-1}$, we start with the graph of $f(x)=2^{x}$ and shift it to the _____ (left/right) 1 unit.

Jeffrey Russell

Jeffrey Russell

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Problem 4

In the formula $A(t)=P\left(1+\frac{r}{n}\right)^{n t}$ for compound interest the
letters $P, r, n,$ and $t$ stand for _____, _____, _____, and _____, respectively, and $A(t)$ stands for _____. So if 100 dollar is invested at an interest rate of $6 \%$ compounded quarterly, then the amount _____ after 2 years is _____.

Jeffrey Russell

Jeffrey Russell

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Problem 5

The exponential function $f(x)=\left(\frac{1}{2}\right)^{x}$ has the _____ asymptote $y=$ _____. This means that as $x \rightarrow \infty,$ we have $\left(\frac{1}{2}\right)^{x} \rightarrow$ _____.

Jeffrey Russell

Jeffrey Russell

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Problem 6

The exponential function $f(x)=\left(\frac{1}{2}\right)^{x}+3$ has the _____ asymptote $y=$ _____. This means that as $x \rightarrow \infty,$ we have $\left(\frac{1}{2}\right)^{x}+3 \rightarrow$ _____.

Jeffrey Russell

Jeffrey Russell

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Problem 7

Evaluating Exponential Functions Use a calculator to evaluate the function at the indicated values. Round your answers to three decimals.
$$
f(x)=4^{x} ; \quad f\left(\frac{1}{2}\right), f(\sqrt{5}), f(-2), f(0.3)
$$

Jeffrey Russell

Jeffrey Russell

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Problem 8

Evaluating Exponential Functions Use a calculator to evaluate the function at the indicated values. Round your answers to three decimals.
$$
f(x)=3^{x-1} ; \quad f\left(\frac{1}{2}\right), f(2.5), f(-1), f\left(\frac{1}{4}\right)
$$

Jeffrey Russell

Jeffrey Russell

Numerade Educator

Problem 9

Evaluating Exponential Functions Use a calculator to evaluate the function at the indicated values. Round your answers to three decimals.
$$
g(x)=\left(\frac{1}{3}\right)^{x+1} ; \quad g\left(\frac{1}{2}\right), g(\sqrt{2}), g(-3.5), g(-1.4)
$$

Jeffrey Russell

Jeffrey Russell

Numerade Educator

Problem 10

Evaluating Exponential Functions Use a calculator to evaluate the function at the indicated values. Round your answers to three decimals.
$$
g(x)=\left(\frac{4}{3}\right)^{3 x} ; \quad g\left(-\frac{1}{2}\right), g(\sqrt{6}), g(-3), g\left(\frac{4}{3}\right)
$$

Jeffrey Russell

Jeffrey Russell

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Problem 11

Graphing Exponential functions Sketch the graph of the function by making a table of values. Use a calculator if necessary.
$$
f(x)=2^{x}
$$

Jeffrey Russell

Jeffrey Russell

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Problem 12

Graphing Exponential functions Sketch the graph of the function by making a table of values. Use a calculator if necessary.
$$
g(x)=8^{x}
$$

Jeffrey Russell

Jeffrey Russell

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Problem 13

Graphing Exponential functions Sketch the graph of the function by making a table of values. Use a calculator if necessary.
$$
f(x)=\left(\frac{1}{3}\right)^{x}
$$

Jeffrey Russell

Jeffrey Russell

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Problem 14

Graphing Exponential functions Sketch the graph of the function by making a table of values. Use a calculator if necessary.
$$
h(x)=(1.1)^{x}
$$

Jeffrey Russell

Jeffrey Russell

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Problem 15

Graphing Exponential functions Sketch the graph of the function by making a table of values. Use a calculator if necessary.
$$
g(x)=3(1.3)^{x}
$$

Jeffrey Russell

Jeffrey Russell

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Problem 16

Graphing Exponential functions Sketch the graph of the function by making a table of values. Use a calculator if necessary.
$$
h(x)=2\left(\frac{1}{4}\right)^{x}
$$

Jeffrey Russell

Jeffrey Russell

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Problem 17

Graphing Exponential Functions Graph both functions on one set of axes.
$$
f(x)=2^{x} \quad \text { and } \quad g(x)=2^{-x}
$$

Jeffrey Russell

Jeffrey Russell

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Problem 18

Graphing Exponential Functions Graph both functions on one set of axes.
$$
f(x)=3^{-x} \text { and } g(x)=\left(\frac{1}{3}\right)^{x}
$$

Jeffrey Russell

Jeffrey Russell

Numerade Educator

Problem 19

Graphing Exponential Functions Graph both functions on one set of axes.
$$
f(x)=4^{x} \quad \text { and } \quad g(x)=7^{x}
$$

Jeffrey Russell

Jeffrey Russell

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Problem 20

Graphing Exponential Functions Graph both functions on one set of axes.
$$
f(x)=\left(\frac{3}{4}\right)^{x} \quad \text { and } \quad g(x)=1.5^{x}
$$

Jeffrey Russell

Jeffrey Russell

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Problem 21

Exponential Functions from a Graph Find the exponential function $f(x)=a^{x}$ whose graph is given.
(Graph cannot copy)

Jeffrey Russell

Jeffrey Russell

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Problem 22

Exponential Functions from a Graph Find the exponential function $f(x)=a^{x}$ whose graph is given.
(Graph cannot copy)

Jeffrey Russell

Jeffrey Russell

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Problem 23

Exponential Functions from a Graph Find the exponential function $f(x)=a^{x}$ whose graph is given.
(Graph cannot copy)

Jeffrey Russell

Jeffrey Russell

Numerade Educator

Problem 24

Exponential Functions from a Graph Find the exponential function $f(x)=a^{x}$ whose graph is given.
(Graph cannot copy)

Jeffrey Russell

Jeffrey Russell

Numerade Educator

Problem 25

Exponential Functions from a Graph Match the exponential function with one of the graphs labeled I or II.
(Graph cannot copy)

Jeffrey Russell

Jeffrey Russell

Numerade Educator

Problem 26

Exponential Functions from a Graph Match the exponential function with one of the graphs labeled I or II.
(Graph cannot copy)

Jeffrey Russell

Jeffrey Russell

Numerade Educator

Problem 27

Graphing Exponential Functions Graph the function, not by plotting points, but by starting from the graphs in Figure $2 .$ State the domain, range, and asymptote.
$$
g(x)=2^{x}-3
$$

Khushbu Rani

Numerade Educator

Problem 28

Graphing Exponential Functions Graph the function, not by plotting points, but by starting from the graphs in Figure $2 .$ State the domain, range, and asymptote.
$$
h(x)=4+\left(\frac{1}{2}\right)^{x}
$$

Jeffrey Russell

Jeffrey Russell

Numerade Educator

Problem 29

Graphing Exponential Functions Graph the function, not by plotting points, but by starting from the graphs in Figure $2 .$ State the domain, range, and asymptote.
$$
f(x)=-3^{x}
$$

Jeffrey Russell

Jeffrey Russell

Numerade Educator

Problem 30

Graphing Exponential Functions Graph the function, not by plotting points, but by starting from the graphs in Figure $2 .$ State the domain, range, and asymptote.
$$
f(x)=10^{-x}
$$

Jeffrey Russell

Jeffrey Russell

Numerade Educator

Problem 31

Graphing Exponential Functions Graph the function, not by plotting points, but by starting from the graphs in Figure $2 .$ State the domain, range, and asymptote.
$$
f(x)=10^{x+3}
$$

Jeffrey Russell

Jeffrey Russell

Numerade Educator

Problem 32

Graphing Exponential Functions Graph the function, not by plotting points, but by starting from the graphs in Figure $2 .$ State the domain, range, and asymptote.
$$
g(x)=2^{x-3}
$$

Jeffrey Russell

Jeffrey Russell

Numerade Educator

Problem 33

Graphing Exponential Functions Graph the function, not by plotting points, but by starting from the graphs in Figure $2 .$ State the domain, range, and asymptote.
$$
y=5^{-x}+1
$$

Jeffrey Russell

Jeffrey Russell

Numerade Educator

Problem 34

Graphing Exponential Functions Graph the function, not by plotting points, but by starting from the graphs in Figure $2 .$ State the domain, range, and asymptote.
$$
h(x)=6-3^{x}
$$

Jeffrey Russell

Jeffrey Russell

Numerade Educator

Problem 35

Graphing Exponential Functions Graph the function, not by plotting points, but by starting from the graphs in Figure $2 .$ State the domain, range, and asymptote.
$$
y=2-\left(\frac{1}{3}\right)^{x}
$$

Jeffrey Russell

Jeffrey Russell

Numerade Educator

Problem 36

Graphing Exponential Functions Graph the function, not by plotting points, but by starting from the graphs in Figure $2 .$ State the domain, range, and asymptote.
$$
y=5^{-x}-3
$$

Jeffrey Russell

Jeffrey Russell

Numerade Educator

Problem 37

Graphing Exponential Functions Graph the function, not by plotting points, but by starting from the graphs in Figure $2 .$ State the domain, range, and asymptote.
$$
h(x)=2^{x-4}+1
$$

Jeffrey Russell

Jeffrey Russell

Numerade Educator

Problem 38

Graphing Exponential Functions Graph the function, not by plotting points, but by starting from the graphs in Figure $2 .$ State the domain, range, and asymptote.
$$
y=3-10^{x-1}
$$

Jeffrey Russell

Jeffrey Russell

Numerade Educator

Problem 39

Graphing Exponential Functions Graph the function, not by plotting points, but by starting from the graphs in Figure $2 .$ State the domain, range, and asymptote.
$$
g(x)=1-3^{-x}
$$

Jeffrey Russell

Jeffrey Russell

Numerade Educator

Problem 40

Graphing Exponential Functions Graph the function, not by plotting points, but by starting from the graphs in Figure $2 .$ State the domain, range, and asymptote.
$$
y=3-\left(\frac{1}{5}\right)^{x}
$$

Jeffrey Russell

Jeffrey Russell

Numerade Educator

Problem 41

Comparing Exponential Functions In these exercises we compare the graphs of two exponential functions.
(a) Sketch the graphs of $f(x)=2^{x}$ and $g(x)=3\left(2^{x}\right)$
(b) How are the graphs related?

Jeffrey Russell

Jeffrey Russell

Numerade Educator

Problem 42

Comparing Exponential Functions In these exercises we compare the graphs of two exponential functions.
(a) Sketch the graphs of $f(x)=9^{x / 2}$ and $g(x)=3^{x}$.
(b) Use the Laws of Exponents to explain the relationship between these graphs.

Jeffrey Russell

Jeffrey Russell

Numerade Educator

Problem 43

Comparing Exponential and Power Functions. Compare the graphs of the power function $f$ and exponential function $g$ by evaluating both of them for $x=0,1,2,3,4,6,8,$ and $10 .$ Then draw the graphs of $f$ and $g$ on the same set of axes.
$$
f(x)=x^{3} ; \quad g(x)=3^{x}
$$

Jeffrey Russell

Jeffrey Russell

Numerade Educator

Problem 44

Comparing Exponential and Power Functions. Compare the graphs of the power function $f$ and exponential function $g$ by evaluating both of them for $x=0,1,2,3,4,6,8,$ and $10 .$ Then draw the graphs of $f$ and $g$ on the same set of axes.
$$
f(x)=x^{4} ; \quad g(x)=4^{x}
$$

Jeffrey Russell

Jeffrey Russell

Numerade Educator

Problem 45

Comparing Exponential and Power Functions In these exercises we use a graphing calculator to compare the rates of growth of the graphs of a power function and an exponential function.
(a) Compare the rates of growth of the functions $f(x)=2^{x}$ and $g(x)=x^{5}$ by drawing the graphs of both functions in the following viewing rectangles.
(i) $[0,5]$ by $[0,20]$
(ii) $[0,25]$ by $\left[0,10^{7}\right]$
(iii) $[0,50]$ by $\left[0,10^{8}\right]$
(b) Find the solutions of the equation $2^{x}=x^{5},$ rounded to one decimal place.

Jeffrey Russell

Jeffrey Russell

Numerade Educator

Problem 46

Comparing Exponential and Power Functions In these exercises we use a graphing calculator to compare the rates of growth of the graphs of a power function and an exponential function.
(a) Compare the rates of growth of the functions $f(x)=3^{x}$ and $g(x)=x^{4}$ by drawing the graphs of both functions in the following viewing rectangles:
(i) $[-4,4]$ by $[0,20]$
(ii) $[0,10]$ by $[0,5000]$
(iii) $[0,20]$ by $\left[0,10^{5}\right]$
(b) Find the solutions of the equation $3^{x}=x^{4},$ rounded to two decimal places.

Jeffrey Russell

Jeffrey Russell

Numerade Educator

Problem 47

Families of Functions Draw graphs of the given family of functions for $c=0.25,0.5,1,2,4 .$ How are the graphs related?
$$
f(x)=c 2^{x}
$$

Jeffrey Russell

Jeffrey Russell

Numerade Educator

Problem 48

Families of Functions Draw graphs of the given family of functions for $c=0.25,0.5,1,2,4 .$ How are the graphs related?
$$
f(x)=2^{c x}
$$

Jeffrey Russell

Jeffrey Russell

Numerade Educator

Problem 49

Getting Information from a Graph Find, rounded to two decimal places, (a) the intervals on which the function is increasing or decreasing and (b) the range of the function.
$$
y=10^{x-x^{2}}
$$

Jeffrey Russell

Jeffrey Russell

Numerade Educator

Problem 50

Getting Information from a Graph Find, rounded to two decimal places, (a) the intervals on which the function is increasing or decreasing and (b) the range of the function.
$$
y=x 2^{x}
$$

Jeffrey Russell

Jeffrey Russell

Numerade Educator

Problem 51

Difference Quotients These exercises involve a difference quotient for an exponential function.
If $f(x)=10^{x}$, show that
$$
\frac{f(x+h)-f(x)}{h}=10^{x}\left(\frac{10^{h}-1}{h}\right)
$$

Jeffrey Russell

Jeffrey Russell

Numerade Educator

Problem 52

Difference Quotients These exercises involve a difference quotient for an exponential function.
If $f(x)=3^{x-1}$, show that
$$
\frac{f(x+h)-f(x)}{h}=3^{x-1}\left(\frac{3^{h}-1}{h}\right)
$$

Jeffrey Russell

Jeffrey Russell

Numerade Educator

Problem 53

Bacteria Growth A bacteria culture contains 1500 bacteria initially and doubles every hour.
(a) Find a function $N$ that models the number of bacteria after $t$ hours.
(b) Find the number of bacteria after 24 hours.

Jeffrey Russell

Jeffrey Russell

Numerade Educator

Problem 54

Mouse Population A certain breed of mouse was introduced onto a small island with an initial population of 320 mice, and scientists estimate that the mouse population is doubling every year.
(a) Find a function $N$ that models the number of mice after t years.
(b) Estimate the mouse population after 8 years.

Jeffrey Russell

Jeffrey Russell

Numerade Educator

Problem 55

Compound Interest An investment of 5000 dollar is deposited into an account in which interest is compounded monthly. Complete the table by filling in the amounts to which the investment grows at the indicated times or interest rates.
$$
r=4 \%
$$

Jeffrey Russell

Jeffrey Russell

Numerade Educator

Problem 56

Compound Interest An investment of 5000 dollar is deposited into an account in which interest is compounded monthly. Complete the table by filling in the amounts to which the investment grows at the indicated times or interest rates.
$$
t=5 \text { years }
$$

Jeffrey Russell

Jeffrey Russell

Numerade Educator

Problem 57

Compound Interest If 10,000 dollar is invested at an interest rate of $3 \%$ per year, compounded semiannually, find the value of the investment after the given number of years.
$$\begin{array}{llll}{\text { (a) } 5 \text { years }} & {\text { (b) } 10 \text { years }} & {\text { (c) } 15 \text { years }}\end{array}$$

Jeffrey Russell

Jeffrey Russell

Numerade Educator

Problem 58

Compound Interest If 2500 dollar is invested at an interest rate of $2.5 \%$ per year, compounded daily, find the value of the investment after the given number of years.
$$\begin{array}{llll}{\text { (a) } 2 \text { years }} & {\text { (b) } 3 \text { years }} & {\text { (c) } 6 \text { years }}\end{array}$$

Jeffrey Russell

Jeffrey Russell

Numerade Educator

Problem 59

Compound Interest If 500 dollar is invested at an interest rate of $3.75 \%$ per year, compounded quarterly, find the value of the investment after the given number of years.
$$\begin{array}{llll}{\text { (a) } 1 \text { year }} & {\text { (b) } 2 \text { years }} & {\text { (c) } 10 \text { years }}\end{array}$$

Jeffrey Russell

Jeffrey Russell

Numerade Educator

Problem 60

Compound Interest If 4000 dollar is borrowed at a rate of $5.75 \%$ interest per year, compounded quarterly, find the amount due at the end of the given number of years.
$$\begin{array}{llll}{\text { (a) } 4 \text { years }} & {\text { (b) } 6 \text { years }} & {\text { (c) } 8 \text { years }}\end{array}$$

Jeffrey Russell

Jeffrey Russell

Numerade Educator

Problem 61

Present Value The present value of a sum of money is the amount that must be invested now, at a given rate of interest, to produce the desired sum at a later date.
Find the present value of 10,000 dollar if interest is paid at a rate of $9 \%$ per year, compounded semiannually, for 3 years.

Jeffrey Russell

Jeffrey Russell

Numerade Educator

Problem 62

Present Value The present value of a sum of money is the amount that must be invested now, at a given rate of interest, to produce the desired sum at a later date.
Find the present value of 100,000 dollar if interest is paid at a rate of $8 \%$ per year, compounded monthly, for 5 years.

Jeffrey Russell

Jeffrey Russell

Numerade Educator

Problem 63

Annual Percentage Yield Find the annual percentage yield for an investment that earns $8 \%$ per year, compounded monthly.

Jeffrey Russell

Jeffrey Russell

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Problem 64

Annual Percentage Yield Find the annual percentage yield for an investment that earns $5 \frac{1}{2} \%$ per year, compounded quarterly.

Jeffrey Russell

Jeffrey Russell

Numerade Educator

Problem 65

Discuss a DiScoveR: Growth of an Exponential Function Suppose you are offered a job that lasts one month, and you are to be very well paid. Which of the following methods of payment is more profitable for you?
(a) One million dollars at the end of the month
(b) Two cents on the first day of the month, 4 cents on the second day, 8 cents on the third day, and, in general, $2^{n}$ cents on the $n$ th day

Jeffrey Russell

Jeffrey Russell

Numerade Educator

Problem 66

Discuss a DISCOVER: The Height of the Graph of an Exponential Function Your mathematics instructor asks
you to sketch a graph of the exponential function
$$
f(x)=2^{x}
$$
for x between 0 and 40, using a scale of 10 units to one inch. What are the dimensions of the sheet of paper you will need to sketch this graph?

Jeffrey Russell

Jeffrey Russell

Numerade Educator