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Applied Calculus for the MLSS A Brief Approach

Soo T. Tan

Chapter 5

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. $3^{-3} \cdot 3^{6}$

Priyanka Sadarangani
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
Priyanka Sadarangani
Numerade Educator
01:11

Problem 3

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

Priyanka Sadarangani
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
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
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
Priyanka Sadarangani
Numerade Educator
02:46

Problem 7

Evaluate the expression.
a. $\frac{5^{3.3} \cdot 5^{-1.6}}{5^{-0.3}}$
b. $\frac{4^{2.7} \cdot 4^{-1.3}}{4^{-0.4}}$

Priyanka Sadarangani
Priyanka Sadarangani
Numerade Educator
02:52

Problem 8

Evaluate the expression.
a. $\left(\frac{1}{16}\right)^{-1 / 4}\left(\frac{27}{64}\right)^{-1 / 3}$
b. $\left(\frac{8}{27}\right)^{-1 / 3}\left(\frac{81}{256}\right)^{-1 / 4}$

Priyanka Sadarangani
Priyanka Sadarangani
Numerade Educator
01:52

Problem 9

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

Priyanka Sadarangani
Priyanka Sadarangani
Numerade Educator
01:34

Problem 10

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
Priyanka Sadarangani
Numerade Educator
01:36

Problem 11

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

Priyanka Sadarangani
Priyanka Sadarangani
Numerade Educator
01:31

Problem 12

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

Priyanka Sadarangani
Priyanka Sadarangani
Numerade Educator
01:49

Problem 13

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

Priyanka Sadarangani
Priyanka Sadarangani
Numerade Educator
01:15

Problem 14

Simplify the expression.
a. $\left(x^{r / s}\right)^{s / r}$
b. $\left(x^{-b / a}\right)^{-a / b}$

Priyanka Sadarangani
Priyanka Sadarangani
Numerade Educator
02:17

Problem 15

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
Priyanka Sadarangani
Numerade Educator
04:34

Problem 16

Simplify the expression.
a. $\frac{\left(a^{m} \cdot a^{-n}\right)^{-2}}{\left(a^{m+n}\right)^{2}}$
b. $\left(\frac{x^{2 n-2} y^{2 n}}{x^{5 n+1} y^{-n}}\right)^{1 / 3}$

Priyanka Sadarangani
Priyanka Sadarangani
Numerade Educator
01:13

Problem 17

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

Priyanka Sadarangani
Priyanka Sadarangani
Numerade Educator
01:15

Problem 18

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

Priyanka Sadarangani
Priyanka Sadarangani
Numerade Educator
01:27

Problem 19

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

Priyanka Sadarangani
Priyanka Sadarangani
Numerade Educator
01:34

Problem 20

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

Priyanka Sadarangani
Priyanka Sadarangani
Numerade Educator
01:04

Problem 21

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

Priyanka Sadarangani
Priyanka Sadarangani
Numerade Educator
01:30

Problem 22

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

Priyanka Sadarangani
Priyanka Sadarangani
Numerade Educator
01:55

Problem 23

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

Priyanka Sadarangani
Priyanka Sadarangani
Numerade Educator
01:40

Problem 24

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

Priyanka Sadarangani
Priyanka Sadarangani
Numerade Educator
01:51

Problem 25

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

Priyanka Sadarangani
Priyanka Sadarangani
Numerade Educator
01:07

Problem 26

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

Priyanka Sadarangani
Priyanka Sadarangani
Numerade Educator
03:11

Problem 27

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

Priyanka Sadarangani
Priyanka Sadarangani
Numerade Educator
03:20

Problem 28

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

Priyanka Sadarangani
Priyanka Sadarangani
Numerade Educator
03:13

Problem 29

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

Priyanka Sadarangani
Priyanka Sadarangani
Numerade Educator
03:30

Problem 30

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

Priyanka Sadarangani
Priyanka Sadarangani
Numerade Educator
03:20

Problem 31

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

Priyanka Sadarangani
Priyanka Sadarangani
Numerade Educator
03:10

Problem 32

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

Priyanka Sadarangani
Priyanka Sadarangani
Numerade Educator
03:53

Problem 33

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

Priyanka Sadarangani
Priyanka Sadarangani
Numerade Educator
03:33

Problem 34

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

Priyanka Sadarangani
Priyanka Sadarangani
Numerade Educator
03:29

Problem 35

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

Priyanka Sadarangani
Priyanka Sadarangani
Numerade Educator
03:33

Problem 36

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

Priyanka Sadarangani
Priyanka Sadarangani
Numerade Educator
01:57

Problem 37

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: $e^{k x}=\left(e^{k}\right)^{x}$

Priyanka Sadarangani
Priyanka Sadarangani
Numerade Educator
02:57

Problem 38

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

Priyanka Sadarangani
Priyanka Sadarangani
Numerade Educator
03:40

Problem 39

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

Priyanka Sadarangani
Priyanka Sadarangani
Numerade Educator
04:50

Problem 40

After having been on the air for more than a decade, Fox's American Idol seemed to be suffering from viewer fatigue. The average number of viewers from the 2011 season through the 2013 season is approximated by $$f(t)=32.744 e^{-0.252 t} \quad(1 \leq t \leq 3)$$ where $f(t)$ is measured in millions, with $t=1$ corresponding to the 2011 season.
a. What was the average number of viewers in the 2011 season?
b. What was the average number of viewers in the 2014 season, assuming that the trend continued into that season?

Suman Saurav Thakur
Suman Saurav Thakur
Numerade Educator
03:02

Problem 41

Developing countries are accelerating the pace of their investment in renewable energy. According to a report by the Frankfurt School of Finance and Management, the amount of investment in renewable energy (in billions of dollars) by developing countries between $2009(t=0)$ and 2012 is given by $$f(t)=64 e^{0.188 t} \quad(0 \leq t \leq 3)$$
a. Find the amount of investment in renewable energy by developing countries in each of the years 2009 through 2012 by completing the following table:
b. Use the information from part (a) to sketch the graph of $f$.

Priyanka Sadarangani
Priyanka Sadarangani
Numerade Educator
04:06

Problem 42

According to a study conducted by Forrester Research, Inc., the amount of money spent by online shoppers in the United States is projected to be $$f(t)=105 e^{0.095 t} \quad(1 \leq t \leq 6)$$ billion dollars in year $t,$ where $t=1$ corresponds to $2011 .$
a. Find the projected spending in each of the years 2011 through 2016 by completing the following table:
$$\begin{array}{cccccccc}\hline t & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\\hline f(t) & & & & & & & \\\hline\end{array}$$
b. Use the information from part (a) to sketch the graph of $f$.

Priyanka Sadarangani
Priyanka Sadarangani
Numerade Educator
03:30

Problem 43

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

Priyanka Sadarangani
Priyanka Sadarangani
Numerade Educator
03:28

Problem 44

The number of cell phone subscribers in the United States between the years 2000 and 2010 is approximated by the function $$N(t)=\frac{385.474}{1+2.52 e^{-0.214 t}} \quad(0 \leq t \leq 10)$$ where $N(t)$ is measured in millions and $t$ is measured in years, with $t=0$ corresponding to the year 2000. How a many cell phone subscribers were there in the United States in 2000 ? If the trend continued, how many subscribers were there in $2012 ?$

Priyanka Sadarangani
Priyanka Sadarangani
Numerade Educator
01:51

Problem 45

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 $$N(t)=\frac{35.5}{1+6.89 e^{-0.8674 t}} \quad(0 \leq t \leq 7)$$ where $N(t)$ is measured in millions and $t$ is measured in years, with $t=0$ corresponding to 2004 What was the projected number of taxpayers subjected to an alternative minimum tax in $2010 ?$

Priyanka Sadarangani
Priyanka Sadarangani
Numerade Educator
05:15

Problem 46

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 per 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 sec?
c. What will be the concentration of the drug in the organ in the long run?
Hint: Evaluate $\lim _{t \rightarrow \infty} x(t)$.
d. Sketch the graph of $x$.

Priyanka Sadarangani
Priyanka Sadarangani
Numerade Educator
05:15

Problem 47

The concentration of a drug in an organ at any time $t$ (in seconds) is given by $$C(t)=\left\{\begin{array}{ll}0.3 t-18\left(1-e^{-t / 60}\right) & \text { if0 } \leq t \leq 20 \\18 e^{-t / 60}-12 e^{-(t-20) / 60} & \text { ift }>20
\end{array}\right.$$ where $C(t)$ is measured in grams per cubic centimeter $\left(g / \mathrm{cm}^{3}\right)$.
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?
d. What will be the concentration of the drug in the long run?
Hint: Evaluate $\lim _{t \rightarrow \infty} C(t)$.

Priyanka Sadarangani
Priyanka Sadarangani
Numerade Educator
02:17

Problem 48

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 dose was taken is given by $$A(t)=\left\{\begin{array}{ll}100 e^{-1.4 t} & \text { if0 } \leq t<1 \\100\left(1+e^{1.4}\right) e^{-1.4 t} & \text { ift } \geq 1\end{array}\right.$$
a. What was the amount of drug in Jane's body immediately after taking the second dose? After 2 days? In the long run?
b. Sketch the graph of $A .$

Adrian Co
Adrian Co
Numerade Educator
01:37

Problem 49

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 $a$ and $b$ are positive numbers, then $(a+b)^{x}=a^{x}+b^{x}$.

Priyanka Sadarangani
Priyanka Sadarangani
Numerade Educator
01:36

Problem 50

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
Priyanka Sadarangani
Numerade Educator
01:58

Problem 51

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
Priyanka Sadarangani
Numerade Educator
01:23

Problem 52

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 $e^{k x}>1,$ then $k>0$ and $x>0$.

Priyanka Sadarangani
Priyanka Sadarangani
Numerade Educator
03:20

Problem 53

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. $f(x)=e^{k x}$ is an increasing function if $k>0$ and a decreasing function if $k<0$.

Priyanka Sadarangani
Priyanka Sadarangani
Numerade Educator
01:51

Problem 54

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. $f(x)=\frac{x}{1+e^{x}}$ is continuous on $(-\infty, \infty)$.

Priyanka Sadarangani
Priyanka Sadarangani
Numerade Educator