College Algebra

James Stewart, Lothar Redlin, Saleem Watson

Chapter 4

Exponential and Logarithmic Functions - all with Video Answers

Educators

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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 $f(6)=$

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Problem 2

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

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Ankit G.

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 _____
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.

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Ankit G.

Numerade Educator

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 ______.

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

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

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Problem 7

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

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Ankit G.

Numerade Educator

Problem 8

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

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Ankit G.

Numerade Educator

Problem 9

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

AG

Ankit G.

Numerade Educator

Problem 10

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

AG

Ankit G.

Numerade Educator

Problem 11

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

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Ankit G.

Numerade Educator

Problem 12

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

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Ankit G.

Numerade Educator

Problem 13

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

AG

Ankit G.

Numerade Educator

Problem 14

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

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Ankit G.

Numerade Educator

Problem 15

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

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Ankit G.

Numerade Educator

Problem 16

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

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Ankit G.

Numerade Educator

Problem 17

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

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Ankit G.

Numerade Educator

Problem 18

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

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Ankit G.

Numerade Educator

Problem 19

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

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Ankit G.

Numerade Educator

Problem 20

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

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Ankit G.

Numerade Educator

Problem 21

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

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Problem 22

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

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Problem 23

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

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Problem 24

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

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Problem 25

Match the exponential function with one of the graphs labeled I or II.
$$f(x)=5^{x+1}$$

AG

Ankit G.

Numerade Educator

Problem 26

Match the exponential function with one of the graphs labeled I or II.
$$f(x)=5^{x}+1$$

AG

Ankit G.

Numerade Educator

Problem 27

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

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Problem 28

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

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Problem 29

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

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Problem 30

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

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Problem 31

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

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Problem 32

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

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Problem 33

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

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Problem 34

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

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Problem 35

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

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Problem 36

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

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Problem 37

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

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Problem 38

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

AG

Ankit G.

Numerade Educator

Problem 39

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

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Problem 40

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

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Problem 41

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?

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Ankit G.

Numerade Educator

Problem 42

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.

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Ankit G.

Numerade Educator

Problem 43

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.
$$f(x)=x^{3} ; \quad g(x)=3^{x}$$

AG

Ankit G.

Numerade Educator

Problem 44

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

AG

Ankit G.

Numerade Educator

Problem 45

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 $\left[0,10^{8}\right]$

AG

Ankit G.

Numerade Educator

Problem 46

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:
$$\text { i. }[-4,4] \text { by }[0,20]$$
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.

AG

Ankit G.

Numerade Educator

Problem 47

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

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Ankit G.

Numerade Educator

Problem 48

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

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Ankit G.

Numerade Educator

Problem 49

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

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Ankit G.

Numerade Educator

Problem 50

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

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Ankit G.

Numerade Educator

Problem 51

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

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Ankit G.

Numerade Educator

Problem 52

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

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Ankit G.

Numerade Educator

Problem 53

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.

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Ankit G.

Numerade Educator

Problem 54

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.

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Problem 55

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.
$$r=4 \%$$
$$\begin{array}{|c|c|}\hline \text { Time (years) } & \text { Amount } \\\hline 1 & \\2 & \\3 & \\4 & \\5 & \\6&\\\hline\end{array}$$

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Ankit G.

Numerade Educator

Problem 56

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.
$t=5$ years
$$\begin{array}{|c|c|}\hline \text { Rate per year } & \text { Amount } \\\hline 1 \% & \\2 \% & \\3 \% & \\4\% & \\5 \% & \\6 \% & \\\hline\end{array}$$

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Ankit G.

Numerade Educator

Problem 57

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.
a. 5 years
b. 10 years
c. 15 years

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Ankit G.

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.
a. 2 years
b. 3 years
c. 6 years

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Ankit G.

Numerade Educator

Problem 59

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.
a. 1 year
b. 2 years
c. 10 years

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Ankit G.

Numerade Educator

Problem 60

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.
a. 4 years
b. 6 years
c. 8 years

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Ankit G.

Numerade Educator

Problem 61

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.

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Problem 62

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.

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Problem 63

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 annual percentage yield for an investment that earns $8 \%$ per year, compounded monthly.

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Ankit G.

Numerade Educator

Problem 64

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

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Ankit G.

Numerade Educator

Problem 65

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

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Ankit G.

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Problem 66

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?

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Ankit G.

Numerade Educator