Algebra and Trigonometry

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 $f(6)=$ _____.

Haley Mortell

Haley Mortell

Numerade Educator

Problem 2

Match the exponential function with one of the graphs
labeled I, II, III, or IV, shown below.
\begin{equation}
\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}
\end{equation}

Mohamed Mohamed

Mohamed Mohamed

Numerade Educator

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 _____ (leftright) 1 unit.

Jeremy Strong

Numerade Educator

Problem 4

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

Mohamed Mohamed

Mohamed Mohamed

Numerade Educator

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 $\left(\frac{1}{2}\right)^{x} \rightarrow$ ______.

Jeremy Strong

Numerade Educator

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 $\left(\frac{1}{2}\right)^{x}+3 \rightarrow$ ______.

Mohamed Mohamed

Mohamed Mohamed

Numerade Educator

Problem 7

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

Haley Mortell

Numerade Educator

Problem 8

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

Mohamed Mohamed

Mohamed Mohamed

Numerade Educator

Problem 9

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

Haley Mortell

Numerade Educator

Problem 10

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

Mohamed Mohamed

Mohamed Mohamed

Numerade Educator

Problem 11

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

Jeremy Strong

Numerade Educator

Problem 12

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

Mohamed Mohamed

Mohamed Mohamed

Numerade Educator

Problem 13

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

Jeremy Strong

Numerade Educator

Problem 14

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

Mohamed Mohamed

Mohamed Mohamed

Numerade Educator

Problem 15

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

Haley Mortell

Numerade Educator

Problem 16

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

Mohamed Mohamed

Mohamed Mohamed

Numerade Educator

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

Numerade Educator

Problem 18

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

Mohamed Mohamed

Mohamed Mohamed

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

Numerade Educator

Problem 20

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

Mohamed Mohamed

Mohamed Mohamed

Numerade Educator

Problem 21

Exponential Functions from a Graph Find the exponential function $f(x)=a^{x}$ whose graph is given.

Mohamed Mohamed

Mohamed Mohamed

Numerade Educator

Problem 22

Exponential Functions from a Graph Find the exponential function $f(x)=a^{x}$ whose graph is given.

Mohamed Mohamed

Mohamed Mohamed

Numerade Educator

Problem 23

Exponential Functions from a Graph Find the exponential function $f(x)=a^{x}$ whose graph is given.

Mohamed Mohamed

Mohamed Mohamed

Numerade Educator

Problem 24

Exponential Functions from a Graph Find the exponential function $f(x)=a^{x}$ whose graph is given.

Mohamed Mohamed

Mohamed Mohamed

Numerade Educator

Problem 25

Exponential Functions from a Graph Match the exponential function with one of the graphs labeled I or II.

Mohamed Mohamed

Mohamed Mohamed

Numerade Educator

Problem 26

Exponential Functions from a Graph Match the exponential function with one of the graphs labeled I or II.

Mohamed Mohamed

Mohamed Mohamed

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.
$$g(x)=2^{x}-3$$

Mohamed Mohamed

Mohamed Mohamed

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.
$$h(x)=4+\left(\frac{1}{2}\right)^{x}$$

Mohamed Mohamed

Mohamed Mohamed

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.
$$f(x)=10^{-x}$$

Mohamed Mohamed

Mohamed Mohamed

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.
$$g(x)=2^{x-3}$$

Mohamed Mohamed

Mohamed Mohamed

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.
$$h(x)=6-3^{x}$$

Mohamed Mohamed

Mohamed Mohamed

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.
$$y=5^{-x}-3$$

Mohamed Mohamed

Mohamed Mohamed

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.
$$y=3-10^{x-1}$$

Mohamed Mohamed

Mohamed Mohamed

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.
$$y=3-\left(\frac{1}{5}\right)^{x}$$

Mohamed Mohamed

Mohamed Mohamed

Numerade Educator

Problem 41

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

Haley Mortell

Numerade Educator

Problem 42

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

Mohamed Mohamed

Mohamed Mohamed

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.
$$f(x)=x^{4} ; \quad g(x)=4^{x}$$

Mohamed Mohamed

Mohamed Mohamed

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.
$$\begin{array}{l}{\text { (a) Compare the rates of growth of the functions } f(x)=2^{x}} \\ {\text { and } g(x)=x^{5} \text { by drawing the graphs of both functions }} \\ {\text { in the following viewing rectangles. }}\end{array}$$
$$\begin{array}{l}{\text { (i) }[0,5] \text { by }[0,20]} \\ {\text { (ii) }[0,25] \text { by }\left[0,10^{7}\right]} \\ {\text { (iii) }[0,50] \text { by }\left[0,10^{8}\right]}\end{array}$$
$$\begin{array}{l}{\text { (b) Find the solutions of the equation } 2^{x}=x^{5}, \text { rounded to }} \\ {\text { one decimal place. }}\end{array}$$

Mohamed Mohamed

Mohamed Mohamed

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.
$$\begin{array}{l}{\text { (a) Compare the rates of growth of the functions } f(x)=3^{x}} \\ {\text { and } g(x)=x^{4} \text { by drawing the graphs of both functions }} \\ {\text { in the following viewing rectangles: }}\end{array}$$
$$\begin{array}{l}{\text { (i) }[-4,4] \text { by }[0,20]} \\ {\text { (ii) }[0,10] \text { by }[0,5000]} \\ {\text { (iii) }[0,20] \text { by }\left[0,10^{5}\right]}\end{array}$$
$$\begin{array}{l}{\text { (b) Find the solutions of the equation } 3^{x}=x^{4}, \text { rounded to }} \\ {\text { two decimal places. }}\end{array}$$

Mohamed Mohamed

Mohamed Mohamed

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?
$$f(x)=c 2^{x}$$

Mohamed Mohamed

Mohamed Mohamed

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?
$$f(x)=2^{c x}$$

Mohamed Mohamed

Mohamed Mohamed

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.
$$y=x 2^{x}$$

Mohamed Mohamed

Mohamed Mohamed

Numerade Educator

Problem 51

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

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
$$\frac{f(x+h)-f(x)}{h}=3^{x-1}\left(\frac{3^{h}-1}{h}\right)$$

Mohamed Mohamed

Mohamed Mohamed

Numerade Educator

Problem 53

Bacteria Growth A bacteria culture contains 1500 bacteria initially and doubles every hour.
$$\begin{array}{l}{\text { (a) Find a function } N \text { that models the number of bacteria }} \\ {\text { after } t \text { hours. }} \\ {\text { (b) Find the number of bacteria after } 24 \text { hours. }}\end{array}$$

Haley Mortell

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.
$$\begin{array}{l}{\text { (a) Find a function } N \text { that models the number of mice after }} \\ {t \text { years. }} \\ {\text { (b) Estimate the mouse population after } 8 \text { years. }}\end{array}$$

Mohamed Mohamed

Mohamed Mohamed

Numerade Educator

Problem 55

Compound Interest An investment of $\$ 5000$ 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.
$$r=4 \%$$

Haley Mortell

Numerade Educator

Problem 56

Compound Interest An investment of $\$ 5000$ 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.
$t=5$ years

Mohamed Mohamed

Mohamed Mohamed

Numerade Educator

Problem 57

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

Haley Mortell

Numerade Educator

Problem 58

Compound Interest If $\$ 2500$ 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.
$${ (a) }2 \text { years } \quad \text { (b) } 3 \text { years } \quad \text { (c) } 6$$

Mohamed Mohamed

Mohamed Mohamed

Numerade Educator

Problem 59

Compound Interest If $\$ 500$ 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.
$$(a)1 \text { year } \quad \text { (b) } 2 \text { years } \quad \text { (c) } 10$$

Haley Mortell

Numerade Educator

Problem 60

Compound Interest If $\$ 4000$ 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.
$$(a) 4 \text { years } \quad \text { (b) } 6 \text { years } \quad \text { (c) } 8$$

Mohamed Mohamed

Mohamed Mohamed

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.
$$\begin{array}{l}{\text { Find the present value of } \$ 10,000 \text { if interest is paid at a }} \\ {\text { rate of } 9 \% \text { per year, compounded semiannually, for }} \\ {3 \text { years. }}\end{array}$$

Haley Mortell

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.
$$\begin{array}{l}{\text { Find the present value of } \$ 100,000 \text { if interest is paid }} \\ {\text { at a rate of } 8 \% \text { per year, compounded monthly, for }} \\ {5 \text { years. }}\end{array}$$

Mohamed Mohamed

Mohamed Mohamed

Numerade Educator

Problem 63

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

Haley Mortell

Numerade Educator

Problem 64

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

Mohamed Mohamed

Mohamed Mohamed

Numerade Educator

Problem 65

DISCUSS = 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?
$$\begin{array}{l}{\text { (a) One million dollars at the end of the month }} \\ {\text { (b) Two cents on the first day of the month, } 4 \text { cents on the }} \\ {\text { second day, } 8 \text { cents on the third day, and, in general, }} \\ {2^{n} \text { cents on the } n \text { th day }}\end{array}$$

Haley Mortell

Numerade Educator

Problem 66

DISCUSS = 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?

Mohamed Mohamed

Mohamed Mohamed

Numerade Educator