Applied Mathematics: For the Managerial, Life, and Social Sciences

Soo T. Tan

Chapter 3 Exponential and Logarithmic Functions - all with Video Answers

Educators

Section 1

Exponential Functions

01:03

Problem 1

Evaluate the expression.
a. $4^{-3} \cdot 4^{5}$
b. $3^{-3}+3^{6}$

Priyanka Sadarangani

Numerade Educator

01:19

Problem 2

Evaluate the expression.
a. $\left(2^{-1}\right)^{3}$
b. $\left(3^{-2}\right)^{3}$

Priyanka Sadarangani

Numerade Educator

01:11

Problem 3

Evaluate the expression.
a. $9(9)^{-1 / 2}$
b. $5(5)^{-1 / 2}$

Priyanka Sadarangani

Numerade Educator

02:07

Problem 4

Evaluate the expression.
a. $\left[\left(-\frac{1}{2}\right)^{3}\right]^{-2}$
b. $\left[\left(-\frac{1}{3}\right)^{2}\right]^{-3}$

Priyanka Sadarangani

Numerade Educator

02:30

Problem 5

Evaluate the expression.
a. $\frac{(-3)^{4}(-3)^{5}}{(-3)^{8}}$
b. $\frac{\left(2^{-4}\right)\left(2^{6}\right)}{2^{-1}}$

Priyanka Sadarangani

Numerade Educator

02:47

Problem 6

Evaluate the expression.
a. $3^{1 / 4} \cdot 9^{-5 / 8}$
b. $2^{3 / 4} \cdot 4^{-3 / 2}$

Priyanka Sadarangani

Numerade Educator

01:52

Problem 7

Simplify the expression.
a. $\left(64 x^{9}\right)^{1 / 3}$
b. $\left(25 x^{3} y^{4}\right)^{1 / 2}$

Priyanka Sadarangani

Numerade Educator

01:34

Problem 8

Simplify the expression.
a. $\left(2 x^{3}\right)\left(-4 x^{-2}\right)$
b. $\left(4 x^{-2}\right)\left(-3 x^{5}\right)$

Priyanka Sadarangani

Numerade Educator

01:36

Problem 9

Simplify the expression.
a. $\frac{6 a^{-5}}{3 a^{-3}}$
b. $\frac{4 b^{-4}}{12 b^{-6}}$

Priyanka Sadarangani

Numerade Educator

01:31

Problem 10

Simplify the expression.
a. $y^{-3 / 2} y^{5 / 3}$
b. $x^{-3 / 5} x^{8 / 3}$

Priyanka Sadarangani

Numerade Educator

01:39

Problem 11

Simplify the expression.
a. $\left(2 x^{3} y^{2}\right)^{3}$
b. $\left(4 x^{2} y^{2} z^{3}\right)^{2}$

Steven Clarke

Numerade Educator

02:17

Problem 12

Simplify the expression.
a. $\frac{5^{0}}{\left(2^{-3} x^{-3} y^{2}\right)^{2}}$
b. $\frac{(x+y)(x-y)}{(x-y)^{0}}$

Priyanka Sadarangani

Numerade Educator

01:13

Problem 13

Solve the equation for $x$.
$$6^{2 x}=6^{4}$$

Priyanka Sadarangani

Numerade Educator

01:15

Problem 14

Solve the equation for $x$.
$$5^{-x}=5^{3}$$

Priyanka Sadarangani

Numerade Educator

01:27

Problem 15

Solve the equation for $x$.
$$3^{3 x-4}=3^{5}$$

Priyanka Sadarangani

Numerade Educator

01:34

Problem 16

Solve the equation for $x$.
$$10^{2 x-1}=10^{x+3}$$

Priyanka Sadarangani

Numerade Educator

01:04

Problem 17

Solve the equation for $x$.
$$(2.1)^{x+2}=(2.1)^{5}$$

Priyanka Sadarangani

Numerade Educator

01:30

Problem 18

Solve the equation for $x$.
$$(-1.3)^{x-2}=(-1.3)^{2 x+1}$$

Priyanka Sadarangani

Numerade Educator

View

Problem 19

Solve the equation for $x$.
$$8^{x}=\left(\frac{1}{32}\right)^{x-2}$$

James Kiss

Numerade Educator

01:40

Problem 20

Solve the equation for $x$.
$$3^{x-x^{2}}=\frac{1}{9^{x}}$$

Priyanka Sadarangani

Numerade Educator

01:51

Problem 21

Solve the equation for $x$.
$$3^{2 x}-12 \cdot 3^{x}+27=0$$

Priyanka Sadarangani

Numerade Educator

01:07

Problem 22

Solve the equation for $x$.
$$2^{2 x}-4 \cdot 2^{x}+4=0$$

Priyanka Sadarangani

Numerade Educator

03:11

Problem 23

Sketch the graphs of the given functions on the same axes.
$y=2^{x}, y=3^{x}$, and $y=4^{x}$

Priyanka Sadarangani

Numerade Educator

03:20

Problem 24

Sketch the graphs of the given functions on the same axes.
$y=\left(\frac{1}{2}\right)^{x}, y=\left(\frac{1}{3}\right)^{x}$, and $y=\left(\frac{1}{4}\right)^{x}$

Priyanka Sadarangani

Numerade Educator

03:13

Problem 25

Sketch the graphs of the given functions on the same axes.
$y=2^{-x}, y=3^{-x}$, and $y=4^{-x}$

Priyanka Sadarangani

Numerade Educator

03:30

Problem 26

Sketch the graphs of the given functions on the same axes.
$y=4^{0.5 x}$ and $y=4^{-0.5 x}$

Priyanka Sadarangani

Numerade Educator

03:20

Problem 27

Sketch the graphs of the given functions on the same axes.
$y=4^{0.5 x}, y=4^{x}$, and $y=4^{2 x}$

Priyanka Sadarangani

Numerade Educator

03:10

Problem 28

Sketch the graphs of the given functions on the same axes.
$y=e^{x}, y=2 e^{x}$, and $y=3 e^{x}$

Priyanka Sadarangani

Numerade Educator

03:53

Problem 29

Sketch the graphs of the given functions on the same axes.
$y=e^{0.5 x}, y=e^{x}$, and $y=e^{1.5 x}$

Priyanka Sadarangani

Numerade Educator

03:33

Problem 30

Sketch the graphs of the given functions on the same axes.
$y=e^{-0.5 x}, y=e^{-x}$, and $y=e^{-1.5 x}$

Priyanka Sadarangani

Numerade Educator

03:29

Problem 31

Sketch the graphs of the given functions on the same axes.
$y=0.5 e^{-x}, y=e^{-x}$, and $y=2 e^{-x}$

Priyanka Sadarangani

Numerade Educator

03:33

Problem 32

Sketch the graphs of the given functions on the same axes.
$y=1-e^{-x}$ and $y=1-e^{-0.5 x}$

Priyanka Sadarangani

Numerade Educator

01:57

Problem 33

A function $f$ has the form $f(x)=A e^{k x}$. Find $f$ if it is known that $f(0)=100$ and $f(1)=120$. Hint: $e^{k t}=\left(e^{k}\right)^{x}$.

Priyanka Sadarangani

Numerade Educator

02:57

Problem 34

If $f(x)=$ Axe $^{-k x}$, find $f(3)$ if $f(1)=5$ and $f(2)=7$. Hint: $e^{k x}=\left(e^{k}\right)^{x}$.

Priyanka Sadarangani

Numerade Educator

03:40

Problem 35

If
$$
f(t)=\frac{1000}{1+B e^{-b t}}
$$
find $f(5)$ given that $f(0)=20$ and $f(2)=30$. Hint: $e^{k n}=\left(e^{k}\right)^{x}$

Priyanka Sadarangani

Numerade Educator

03:01

Employers are increasingly turning to GPS (global positioning system) technology to keep track of their fleet vehicles. The estimated number of automatic vehicle trackers installed on fleet vehicles in the United States is approximated by
$$
N(t)=0.6 e^{0.17 t} \quad(0 \leq t \leq 5)
$$
where $N(t)$ is measured in millions and $t$ is measured in years, with $t=0$ corresponding to 2000 .
a. What was the number of automatic vehicle trackers installed in the year $2000 ?$ How many were projected to be installed in $2005 ?$
b. Sketch the graph of $N$.

James Kiss

Numerade Educator

10:42

Problem 37

Because of medical technology advances, the disability rates for people over $65 \mathrm{yr}$ old have been dropping rather dramatically. The function
$$
R(t)=26.3 e^{-0.016} \quad(0 \leq t \leq 18)
$$
gives the disability rate $R(t)$, in percent, for people over age 65 from $1982(t=0)$ through 2000 , where $t$ is measured in years.
a. What was the disability rate in $1982 ?$ In $1986 ?$ In 1994 ? In 2000 ?
b. Sketch the graph of $R$.

Oswaldo Jiménez

Numerade Educator

04:39

Problem 38

The percentage of families that were married households between 1970 and 2000 is approximately
$$
P(t)=86.9 e^{-0.05 t} \quad(0 \leq t \leq 3)
$$
where $t$ is measured in decades, with $t=0$ corresponding to the beginning of 1970 .
a. What percentage of families were married households at the beginning of $1970,1980,1990$, and 2000 ?
b. Sketch the graph of $P$.

Stark Ledbetter

Numerade Educator

01:39

Problem 39

According to a study conducted in 2000 , the projected number of Web addresses (in billions) is approximated by the function
$$
N(t)=0.45 e^{0.5696} \quad(0 \leq t \leq 5)
$$
where $t$ is measured in years, with $t=0$ corresponding to the beginning of 1997 .
a. Complete the following table by finding the number of Web addresses in each year:
b. Sketch the graph of $N$.

Nick Johnson

Numerade Educator

03:30

Problem 40

The number of Internet users in China is projected to be
$$
N(t)=94.5 e^{0.2 t} \quad(1 \leq t \leq 6)
$$
where $N(t)$ is measured in millions and $t$ is measured in years, with $t=1$ corresponding to the beginning of 2005 .
a. How many Internet users were there at the beginning of $2005 ?$ At the beginning of 2006 ?
b. How many Internet users are there expected to be at the beginning of 2010 ?
c. Sketch the graph of $N$.

Priyanka Sadarangani

Numerade Educator

01:51

Problem 41

The alternative minimum tax was created in 1969 to prevent the very wealthy from using creative deductions and shelters to avoid having to pay anything to the Internal Revenue Service. But it has increasingly hit the middle class. The number of taxpayers subjected to an alternative minimum tax is projected to be
$$
N(t)=\frac{35.5}{1+6.89 e^{-0.8674 t}} \quad(0 \leq t \leq 6)
$$
where $N(t)$ is measured in millions and $t$ is measured in years, with $t=0$ corresponding to 2004 . What is the projected number of taxpayers subjected to an alternative minimum tax in 2010 ?

Priyanka Sadarangani

Numerade Educator

05:15

Problem 42

The concentration of a drug in an organ at any time $t$ (in seconds) is given by
$$
C(t)=\left\{\begin{array}{ll}
0.3 t-18\left(1-e^{-260}\right) & \text { if } 0 \leq t \leq 20 \\
18 e^{-560}-12 e^{-(t-20) \sqrt{6}} & \text { if } t>20
\end{array}\right.
$$
where $C(t)$ is measured in grams/cubic centimeter $\left(\mathrm{g} / \mathrm{cm}^{3}\right)$.
a. What is the initial concentration of the drug in the organ?
b. What is the concentration of the drug in the organ after 10 sec?
c. What is the concentration of the drug in the organ after 30 sec?

Priyanka Sadarangani

Numerade Educator

04:58

Problem 43

The concentration of a drug in an organ at any time $t$ (in seconds) is given by
$$
x(t)=0.08+0.12\left(1-e^{-0.02 t}\right)
$$
where $x(t)$ is measured in grams/cubic centimeter $\left(\mathrm{g} / \mathrm{cm}^{3}\right)$.
a. What is the initial concentration of the drug in the organ?
b. What is the concentration of the drug in the organ after $20 \mathrm{sec} ?$

Priyanka Sadarangani

Numerade Educator

02:17

Problem 44

Jane took $100 \mathrm{mg}$ of a drug in the morning and another $100 \mathrm{mg}$ of the same drug at the same time the following morning. The amount of the drug in her body $t$ days after the first dosage was taken is given by
$$
A(t)=\left\{\begin{array}{ll}
100 e^{-1.4 t} & \text { if } 0 \leq t<1 \\
100\left(1+e^{1.4}\right) e^{-1 . A r} & \text { if } t \geq 1
\end{array}\right.
$$
What was the amount of drug in Jane's body immediately after taking the second dose? After 2 days?

Adrian Co

Numerade Educator

02:02

Problem 45

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
$$\left(x^{2}+1\right)^{3}=x^{6}+1$$

Gregory Higby

Numerade Educator

01:36

Problem 46

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
$$e^{x y}=e^{x} e^{y}$$

Priyanka Sadarangani

Numerade Educator

01:36

Problem 47

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
If $x<y$, then $e^{x}<e^{y}$.

Priyanka Sadarangani

Numerade Educator

01:58

Problem 48

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
If $0<b<1$ and $x<y$, then $b^{x}>b^{y}$.

Priyanka Sadarangani

Numerade Educator

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