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Trigonometry: Student Study and Solutions Manual

Ron Larson

Chapter 5

Exponential and Logarithmic Functions - all with Video Answers

Educators

Section 1

Review Exercises

01:04

Problem 1

$$
\begin{aligned}
f(x) & =0.3^x \\
f(1.5) & =0.3^{15}=0.164
\end{aligned}
$$

Kian Manafi
Kian Manafi
Numerade Educator
02:01

Problem 3

$$
\begin{aligned}
& f(x)=2^{-0.5 x} \\
& f(\pi)=2^{-0.5(x)} \approx 0.337
\end{aligned}
$$

Manik Pulyani
Manik Pulyani
Numerade Educator
01:22

Problem 5

$$
\begin{aligned}
f(x) & =7\left(0.2^x\right) \\
f(-\sqrt{11}) & =7\left(0.2^{-\sqrt{11}}\right) \\
& \approx 1456.529
\end{aligned}
$$

H M
H M
Numerade Educator
01:02

Problem 7

$f(x)=5^x, g(x)=5^x+1$
Because $g(x)=f(x)+1$, the graph of $g$ can be obtained by shifting the graph of fone unit upward.

Christopher Stanley
Christopher Stanley
Numerade Educator
06:06

Problem 9

$f(x)=3^x, g(x)=1-3^x$
Because $g(x)=1-f(x)$, the graph of $g$ can be obtained by reflecting the graph of $f$ in the $x$-axis and shifting the graph one unit upward. (Note: This is equivalent to shifting the graph of $f$ one unit upward and then reflecting the graph in the $x$-axis.)

Ryan Mcalister
Ryan Mcalister
Numerade Educator
00:33

Problem 10

$f(x)=4^{-x}+4$
Horizontal asymptote: $y=4$
$$
\begin{array}{|l|c|c|c|c|c|}
\hline x & -1 & 0 & 1 & 2 & 3 \\
\hline f(x) & 8 & 5 & 4.25 & 4.063 & 4.016 \\
\hline
\end{array}
$$
(FIGURE CAN'T COPY)

James Kiss
James Kiss
Numerade Educator
01:26

Problem 13

$f(x)=5^{x-2}+4$
Horizontal asymptote: $y=4$
$$
\begin{array}{|l|l|l|l|l|l|}
\hline x & -1 & 0 & 1 & 2 & 3 \\
\hline f(x) & 4.008 & 4.04 & 4.2 & 5 & 9 \\
\hline
\end{array}
$$
(FIGURE CAN'T COPY)

Christopher Stanley
Christopher Stanley
Numerade Educator
01:05

Problem 15

$f(x)=\left(\frac{1}{2}\right)^{-2}+3=2^x+3$
Horizontal asymptote: $y=3$
$$
\begin{array}{|l|c|c|c|c|c|}
\hline x & -2 & -1 & 0 & 1 & 2 \\
\hline f(x) & 3.25 & 3.5 & 4 & 5 & 7 \\
\hline
\end{array}
$$
(FIGURE CAN'T COPY)

Jeffrey Russell
Jeffrey Russell
Numerade Educator
01:23

Problem 17

$$
\begin{aligned}
\left(\frac{1}{3}\right)^{x-3} & =9 \\
\left(\frac{1}{3}\right)^{x-3} & =3^2 \\
\left(\frac{1}{3}\right)^{x-3} & =\left(\frac{1}{3}\right)^{-2} \\
x-3 & =-2 \\
x & =1
\end{aligned}
$$

Rukhmani Jain
Rukhmani Jain
Numerade Educator
View

Problem 19

$$
\begin{aligned}
e^{3 x-5} & =e^7 \\
3 x-5 & =7 \\
3 x & =12 \\
x & =4
\end{aligned}
$$

Nicole Hoffman
Nicole Hoffman
Numerade Educator
00:13

Problem 21

$e^1 \approx 2980.958$

Amy Jiang
Amy Jiang
Numerade Educator
View

Problem 23

$e^{-1.7} \approx 0.183$

Nick Johnson
Nick Johnson
Numerade Educator
02:16

Problem 25

$h(x)=e^{-x / 2}$
$$
\begin{array}{|l|c|c|c|c|c|}
\hline x & -2 & -1 & 0 & 1 & 2 \\
\hline h(x) & 2.72 & 1.65 & 1 & 0.61 & 0.37 \\
\hline
\end{array}
$$
(FIGURE CAN'T COPY)

Joseph Lentino
Joseph Lentino
Numerade Educator
02:12

Problem 27

$f(x)=e^{x+2}$
$$
\begin{array}{|l|c|c|c|c|c|}
\hline x & -3 & -2 & -1 & 0 & 1 \\
\hline f(x) & 0.37 & 1 & 2.72 & 7.39 & 20.09 \\
\hline
\end{array}
$$
(FIGURE CAN'T COPY)

Joseph Lentino
Joseph Lentino
Numerade Educator
03:18

Problem 29

$F(t)=1-e^{-1 / p}$
(a) $F\left(\frac{1}{2}\right) \approx 0.154$
(b) $F(2)=0.487$
(c) $F(5)=0.811$

Yingtai Xiao
Yingtai Xiao
Numerade Educator
01:55

Problem 31

$$P=\$ 5000$$, $r=3 \%$, $t=10$ years Compounded $n$ times per year: $A=P\left(1+\frac{r}{n}\right)^*=5000\left(1+\frac{0.03}{n}\right)^{\text {tt. }}$ Compounded continuously: $A=P e^n=5000 e^{a m(10)}$
$$
\begin{array}{|c|c|c|c|c|c|c|}
\hline n & 1 & 2 & 4 & 12 & 365 & \text { Continuous } \\
\hline A & 56719.58 & \$ 6734.28 & \$ 6741.74 & \$ 6746.77 & \$ 6749.21 & \$ 6749.29 \\
\hline
\end{array}
$$

Monica Miller
Monica Miller
Numerade Educator
00:57

Problem 33

$$
\begin{aligned}
3^1 & =27 \\
\log _3 27 & =3
\end{aligned}
$$

Dwijendra Rao
Dwijendra Rao
Numerade Educator
02:04

Problem 35

$$
\begin{aligned}
e^{0.13} & =2.2255 \ldots \\
\ln 2.2255 \ldots & =0.8
\end{aligned}
$$

Puneet Prajapati
Puneet Prajapati
Numerade Educator
01:49

Problem 37

$$
\begin{aligned}
f(x) & =\log x \\
f(1000) & =\log 1000 \\
& =\log 10^3=3
\end{aligned}
$$

Sherrie Fenner
Sherrie Fenner
Numerade Educator
00:55

Problem 39

$$
\begin{aligned}
g(x) & =\log _2 x \\
g\left(\frac{1}{4}\right) & =\log _2 \frac{1}{4} \\
& =\log _2 2^{-2}=-2
\end{aligned}
$$

AG
Ankit Gupta
Numerade Educator
01:29

Problem 41

$$
\begin{aligned}
\log _4(x+7) & =\log _4 14 \\
x+7 & =14 \\
x & =7
\end{aligned}
$$

Heather Zimmers
Heather Zimmers
Numerade Educator
01:24

Problem 43

$$
\begin{aligned}
\ln (x+9) & =\ln 4 \\
x+9 & =4 \\
x & =-5
\end{aligned}
$$

Nishant Tyagi
Nishant Tyagi
Numerade Educator
05:10

Problem 45

$g(x)=\log _7 x \Rightarrow x=7_y$
Domain: $(0, \infty)$
$x$-intercept: $(1,0)$
Vertical asymptote: $x=0$
$$
\begin{array}{|l|c|l|l|l|}
\hline x & \frac{1}{7} & 1 & 7 & 49 \\
\hline g(x) & -1 & 0 & 1 & 2 \\
\hline
\end{array}
$$
(FIGURE CAN'T COPY)

Emily Keffer
Emily Keffer
Numerade Educator
03:38

Problem 47

$f(x)=4-\log (x+5)$
Domain: $(-5, \infty)$
Because
$$
\begin{aligned}
4-\log (x+5)=0 \Rightarrow \log (x+5) & =4 \\
x+5 & =10^4 \\
x & =10^4-5 \\
& =9995 .
\end{aligned}
$$
$x$-intercept: $(9995,0)$
Vertical asymptote; $x=-5$
$$
\begin{array}{|l|c|c|c|c|c|c|}
\hline x & -4 & -3 & -2 & -1 & 0 & 1 \\
\hline f(x) & 4 & 3.70 & 3.52 & 3.40 & 3.30 & 3.22 \\
\hline
\end{array}
$$
(FIGURE CAN'T COPY)

Jillian Rae Villa
Jillian Rae Villa
Numerade Educator
00:46

Problem 49

$f(22.6)=\ln 22.6=3.118$

Amy Jiang
Amy Jiang
Numerade Educator
01:17

Problem 51

$f(\sqrt{e})=\frac{1}{2} \ln \sqrt{e}=0.25$

Yingtai Xiao
Yingtai Xiao
Numerade Educator
01:17

Problem 53

$f(x)=\ln x+3$
Domain: $(0, \infty)$
$$
\begin{aligned}
\ln x+3 & =0 \\
\ln x & =-3 \\
x & =e^{-3}
\end{aligned}
$$
$x$-intercept: $\left(e^{-3}, 0\right)$
Vertical asymptote: $x=0$
$$
\begin{array}{|l|c|c|c|c|c|}
\hline x & 1 & 2 & 3 & \frac{1}{2} & \frac{1}{4} \\
\hline f(x) & 3 & 3.69 & 4.10 & 2.31 & 1.61 \\
\hline
\end{array}
$$

AG
Ankit Gupta
Numerade Educator
00:48

Problem 55

$h(x)=\ln \left(x^2\right)=2 \ln |x|$
Domain: $(-\infty, 0) \cup(0, \infty)$
$x$-intercepts: $( \pm 1,0)$
Vertical asymplote: $x=0$
$$
\begin{array}{|l|l|l|l|l|l|}
\hline x & \pm 0.5 & \pm 1 & \pm 2 & \pm 3 & \pm 4 \\
\hline h(x) & -1.39 & 0 & 1.39 & 2.20 & 2.77 \\
\hline
\end{array}
$$
(FIGURE CAN'T COPY)

Nick Johnson
Nick Johnson
Numerade Educator
02:41

Problem 57

$$
\begin{aligned}
h & =116 \log (a+40)-176 \\
h(55) & =116 \log (55+40)-176 \\
& \approx 53.4 \text { inches }
\end{aligned}
$$

Vysakh M
Vysakh M
Numerade Educator
04:51

Problem 59

(a) $\log _2 6=\frac{\log 6}{\log 2} \approx 2.585$
(b) $\log _2 6=\frac{\ln 6}{\ln 2} \approx 2.585$

Harshita Goel
Harshita Goel
Numerade Educator
01:17

Problem 61

(a) $\log _{\sqrt{2}} 5=\frac{\log 5}{\log (1 / 2)}=-2.322$
(b) $\log _{\sqrt{2}} 5=\frac{\ln 5}{\ln (1 / 2)}=-2.322$

Swati Agarwal
Swati Agarwal
Numerade Educator
00:09

Problem 63

$$
\begin{aligned}
\log 18 & =\log \left(2 \cdot 3^2\right) \\
& =\log 2+2 \log 3 \\
& \approx 1.255
\end{aligned}
$$

AG
Ankit Gupta
Numerade Educator
00:52

Problem 65

$$
\text { } \begin{aligned}
\ln 20 & =\ln \left(2^2 \cdot 5\right) \\
& =2 \ln 2+\ln 5=2.996
\end{aligned}
$$

Linh Vu
Linh Vu
Numerade Educator
01:05

Problem 67

$$
\begin{aligned}
\log _5 5 x^2 & =\log _5 5+\log _5 x^2 \\
& =1+2 \log _5 x
\end{aligned}
$$

Aman Gupta
Aman Gupta
Numerade Educator
03:37

Problem 69

$\log _3 \frac{9}{\sqrt{x}}=\log _3 9-\log _3 \sqrt{x}$
$$
\begin{aligned}
& =\log _1 3^2-\log _3 x^{1 / 2} \\
& =2-\frac{1}{2} \log _3 x
\end{aligned}
$$

Shima Shaw
Shima Shaw
Numerade Educator

Problem 71

$\ln x^2 y^2 z=\ln x^2+\ln y^2+\ln =$
$$
=2 \ln x+2 \ln y+\ln z
$$

Check back soon!

Problem 73

$\log _2 5+\log _2 x=\log _2 5 x$

Check back soon!

Problem 75

$\ln x-\frac{1}{4} \ln y=\ln x-\ln \sqrt[4]{y}=\ln \frac{x}{\sqrt[4]{y}}$

Check back soon!
03:43

Problem 77

$$
\begin{aligned}
\frac{1}{2} \log _3 x-2 \log _3(y+8) & =\log _3 x^{y 2}-\log _3(y+8)^2 \\
& =\log _3 \sqrt{x}-\log _3(y+8)^2 \\
& =\log _5 \frac{\sqrt{x}}{(y+8)^2}
\end{aligned}
$$

Anas Venkitta
Anas Venkitta
Numerade Educator
01:18

Problem 79

$t=50 \log \frac{18,000}{18,000-h}$
(a) Domain: $0 \leq h<18,000$
(b) (FIGURE CAN'T COPY)
Vertical asymptote: $h=18,000$
(c) As the plane approaches its absolute ceiling, it climbs at a slower rate, so the time required increases.
(d) $50 \log \frac{18,000}{18,000-4000}=5.46$ minutes

Carson Merrill
Carson Merrill
Numerade Educator
01:02

Problem 81

$$
\begin{aligned}
& 5^x=125 \\
& 5^x=5^3 \\
& x=3
\end{aligned}
$$

AG
Ankit Gupta
Numerade Educator
00:34

Problem 83

$$
\begin{aligned}
& e^2=3 \\
& x=\ln 3 \approx 1.099
\end{aligned}
$$

K B
K B
Numerade Educator
00:33

Problem 85

$$
\begin{aligned}
\ln x & =4 \\
x & =e^4 \approx 54.598
\end{aligned}
$$

Heather Zimmers
Heather Zimmers
Numerade Educator
01:03

Problem 87

$$
\begin{aligned}
e^{4 x} & =e^{x^2+1} \\
4 x & =x^2+3 \\
0 & =x^2-4 x+3 \\
0 & =(x-1)(x-3) \\
x & =1, x=3
\end{aligned}
$$

Anurag Kumar
Anurag Kumar
Numerade Educator
00:51

Problem 89

$2^x-3=29$
$$
\begin{aligned}
2^x & =32 \\
2^x & =2^5 \\
x & =5
\end{aligned}
$$

Isabelle Maxwell
Isabelle Maxwell
Numerade Educator
00:40

Problem 91

$25 e^{-43 r}=12$
Graph $y_1=25 e^{-0.2 x}$ and $y_2=12$.
(FIGURE CAN'T COPY)
The graphs intersect at $x \approx 2.447$.

AG
Ankit Gupta
Numerade Educator
00:14

Problem 93

$$
\begin{aligned}
\ln 3 x & =8.2 \\
e^{\operatorname{ls} 3 x} & =e^{8.2} \\
3 x & =e^{x .2} \\
x & =\frac{e^{k .2}}{3} \approx 1213.650
\end{aligned}
$$

Vivek Kumar
Vivek Kumar
Numerade Educator
00:43

Problem 95

$\ln x-\ln 3=2$
$$
\begin{aligned}
\ln \frac{x}{3} & =2 \\
e^{\ln (())} & =e^2 \\
\frac{x}{3} & =e^2 \\
x & =3 e^2 \approx 22.167
\end{aligned}
$$

Heather Zimmers
Heather Zimmers
Numerade Educator
01:51

Problem 97

$$
\begin{aligned}
\log _s(x-1) & =\log _x(x-2)-\log _s(x+2) \\
\log _s(x-1) & =\log _s\left(\frac{x-2}{x+2}\right) \\
x-1 & =\frac{x-2}{x+2} \\
(x-1)(x+2) & =x-2 \\
x^2+x-2 & =x-2 \\
x^2 & =0 \\
x & =0
\end{aligned}
$$
Because $x=0$ is not in the domain of $\log _x(x-1)$ or of $\log _s(x-2)$, it is an extraneous solution. The cquation has no solution.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
00:50

Problem 99

$$
\text { } \begin{aligned}
\log (1-x) & =-1 \\
1-x & =10^{-1} \\
1-\frac{1}{10} & =x \\
x & =0.900
\end{aligned}
$$

Tani Iqbal
Tani Iqbal
Numerade Educator

Problem 101

$2 \ln (x+3)-3=0$
Graph $y=2 \ln (x+3)-3$.
The $x$-intercept is at $x=1.482$.
(FIGURE CAN'T COPY)

Check back soon!
00:31

Problem 103

$6 \log \left(x^2+1\right)-x=0$
Graph $y_1=6 \log \left(x^2+1\right)-x$.
The $x$-intercepts are at $x=0, x=0.416$, and $x \approx 13.627$.
(FIGURE CAN'T COPY)

Trinity Steen
Trinity Steen
Numerade Educator
03:33

Problem 105

$P=8500, A=3(8500)=25,500, r=3.5 \%$
$$
\begin{aligned}
A & =P e^t \\
25,500 & =8500 e^{\text {ents }} \\
3 & =e^{2012 s} \\
\ln 3 & =0.035 t \\
t & =\frac{\ln 3}{0.035} \approx 31.4 \text { years }
\end{aligned}
$$

Christopher Stanley
Christopher Stanley
Numerade Educator
02:10

Problem 107

$y=3 e^{-2 / 3}$
Exponential decay model
Matches graph (e).
108. $y=4 e^{2 . p}$
Exponential growth model
Matches graph (b).

Chengyu Li
Chengyu Li
Numerade Educator
00:37

Problem 109

$y=\ln (x+3)$
Logarithmic model
Vertical asymptote: $x=-3$
Graph includes $(-2,0)$
Matches graph (f).

Kara Merfeld
Kara Merfeld
Numerade Educator
01:30

Problem 110

$y=7-\log (x+3)$
Logarithmic model
Vertical asymptote: $x=-3$
Matches graph (d).

AG
Ankit Gupta
Numerade Educator
03:34

Problem 111

$y=2 e^{-(x+4)^2 / \beta}$
Gaussian model
Matches graph (a).

AG
Ankit Gupta
Numerade Educator
02:10

Problem 112

$y=\frac{6}{1+2 e^{-2 x}}$
Logistics growth model
Matches graph (c).

Chengyu Li
Chengyu Li
Numerade Educator
03:13

Problem 113

$y=a e^{\text {hn }}$
Using the point $(0,2)$, you have
$$
\begin{aligned}
2 & =a e^{(9)} \\
2 & =a e^2 \\
2 & =a(1) \\
2 & =a
\end{aligned}
$$
Then, using the point $(4,3)$, you have
$$
\begin{aligned}
& 3=2 e^{M(4)} \\
& 3=2 e^{4 t} \\
& \frac{1}{2}=e^{4 t} \\
& \ln \frac{3}{2}=4 b \\
& \frac{1}{4} \ln \left(\frac{3}{2}\right)=b
\end{aligned}
$$
So, $y=2 e^{\frac{1}{4}=\left(\frac{1}{2}\right)}$
or
$$
y=2 e^{e . \text {.644t }}
$$

Amy Jiang
Amy Jiang
Numerade Educator
01:14

Problem 115

$y=0.0499 e^{-(x-7)^2 / 28}, 40 \leq x \leq 100$
Graph $y_1=0.0499 e^{-(x-1)^2 / 2 z}$.
(FIGURE CAN'T COPY)
The average test score is 71 .

Kimberly Waterbury
Kimberly Waterbury
Numerade Educator
02:12

Problem 117

$\quad \beta=10 \log \left(\frac{I}{10^{-12}}\right)$
(a)
$$
\begin{aligned}
& \beta=60 \\
& I=10^{\text {Npho-12 }} \\
& I=10^{135}{ }^{13-12} \\
& =10^{15} \\
& =10 \sqrt{10} \text { watts } / \mathrm{m}^2 \\
& =10^{-6} \text { Watt } / \mathrm{m}^2 \\
&
\end{aligned}
$$
(b) $\beta=135$
(c) $\beta=1$
$$
\frac{\beta}{10}=\log \left(\frac{l}{10^{-12}}\right)
$$
$$
\begin{aligned}
I & =10^{\text {ท }_{e-12}} \\
& =10^{\frac{1}{10}} \times 10^{-12}
\end{aligned}
$$
$$
10^{n / 10}=\frac{t}{10^{-12}}
$$
$$
I=10^{\mathrm{ph0}-12}
$$
$\approx 1.259 \times 10^{-2}$ watt $/ \mathrm{m}^2$

Heather Zimmers
Heather Zimmers
Numerade Educator
00:55

Problem 119

True. By the inverse properties, $\log _3 b^{2 r}=2 x$.

Dwijendra Rao
Dwijendra Rao
Numerade Educator