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13.1.10: Chapter 10

( \newcommand{\kernel}{\mathrm{null}\,}\)

Be Prepared

10.1

f(4)=; g(f(4))=

10.2

x=23y+4

10.3

x

10.4

x

10.5

1; 1

10.6

12; 3

10.7

x=9,x=9

10.8

19

10.9

x=7

10.10

1; a

10.11

(x2y)13

10.12

2.565

10.13

x=4,x=4

10.14

x=2,x=3

10.15

x=5,x=1

Try It

10.1

15x+1 15x9
15x27x2

10.2

24x23 24x23
24x238x+15

10.3

–8 5 40

10.4

65 10 5

10.5

One-to-one function
Function; not one-to-one

10.6

Not a function
Function; not one-to-one

10.7

Not a function One-to-one function

10.8

Function; not one-to-one One-to-one function

10.9

Inverse function: {(4,0),(7,1),(10,2),(13,3)}. Domain: {4,7,10,13}. Range: {0,1,2,3}.

10.10

Inverse function: {(4,−1),(1,−2),(0,−3),(2,−4)}. Domain: {0,1,2,4}. Range: {−4,−3,−2,−1}.

10.11
This figure shows a line from (negative 4, negative 3) to (negative 2, negative 2) then to (negative 1, 0) then to (2, 1) and then to (3, 4).
10.12
Graph extends from negative 4 to 4 on both axes. Points plotted are (negative 3, 4), (0, 3), (1, 2), and (4, 1). Line segments connect points.
10.13

g(f(x))=x, and f(g(x))=x, so they are inverses.

10.14

g(f(x))=x, and f(g(x))=x, so they are inverses.

10.15

f−1(x)=x+35

10.16

f−1(x)=x58

10.17

f−1(x)=x5+23

10.18

f−1(x)=x4+76

10.19 This figure shows a curve that slopes swiftly upward from just above (negative 3, 0) through (0, 1) up to (1, 4).
10.20 This figure shows a curve that slopes swiftly upward from just above (negative 3, 0) through (0, 1) up to (1, 5).
10.21 This figure shows a curve that passes through (negative 1, 4), (0, 1) to a point just above (3, 0).
10.22 This figure shows a curve that passes through (negative 1, 5), (0, 1) to a point just above (3, 0).
10.23 This figure shows the graphs of two functions. The first function f of x equals 2 to the x power is marked in blue and corresponds to a curve that passes through the points (negative 1, 1 over 2), (0, 1) and (1, 2). The second function g of x equals 2 to the x minus 1 power is marked in red and passes through the points (0, 1 over 2), (1, 1), and (2, 2).
10.24 This figure shows the graphs of two functions. The first function f of x equals 3 to the x power is marked in blue and corresponds to a curve that passes through the points (negative 1, 1 over 3), (0, 1) and (1, 3). The second function g of x equals 3 to the x plus 1 power is marked in red and passes through the points (negative 2, 1 over 3), (negative 1, 1), and (0, 3).
10.25 This figure shows the graphs of two functions. The first function f of x equals 3 to the x power is marked in blue and corresponds to a curve that passes through the points (negative 1, 1 over 3), (0, 1) and (1, 3). The second function g of x equals 3 to the x power plus 2 is marked in red and passes through the points (negative 1, 7 over 3), (0, 3) and (1, 5).
10.26 This figure shows the graphs of two functions. The first function f of x equals 4 to the x power is marked in blue and corresponds to a curve that passes through the points (negative 1, 1 over 4), (0, 1) and (1, 4). The second function g of x equals 4 to the x power minus 2 is marked in red and passes through the points (negative 1, negative 7 over 4), (0, negative 1), and (1, 2).
10.27

x=2

10.28

x=4

10.29

x=−1,x=2

10.30

x=−2,x=3

10.31

$22,332.96
$22,362.49 $22,377.37

10.32

$21,071.81 $21,137.04
$21,170.00

10.33

She will find 166 bacteria.

10.34

They will find 1,102 viruses.

10.35

log39=2
log77=12 log13127=x

10.36

log464=3
log434=13 log12132=x

10.37

64=43
1=x0 1100=10−2

10.38

27=33 1=30
110=10−1

10.39


x=8 x=125 x=2

10.40



x=9 x=243 x=3

10.41


2 12 −5

10.42

2 13 −2

10.43 This figure shows the logarithmic curve going through the points (1 over 3, negative 1), (1, 0), and (3, 1).
10.44 This figure shows the logarithmic curve going through the points (1 over 5, negative 1), (1, 0), and (5, 1).
10.45 This figure shows the logarithmic curve going through the points (1 over 2, 1), (1, 0), and (2, negative 1).
10.46 This figure shows the logarithmic curve going through the points (1 over 4, 1), (1, 0), and (4, negative 1).
10.47



a=11
x=e7

10.48



a=4
x=e9

10.49



x=13
x=2

10.50



x=6
x=1

10.51

The quiet dishwashers have a decibel level of 50 dB.

10.52

The decibel level of heavy traffic is 90 dB.

10.53

The intensity of the 1906 earthquake was about 8 times the intensity of the 1989 earthquake.

10.54

The intensity of the earthquake in Chile was about 1,259 times the intensity of the earthquake in Los Angeles.

10.55

0 1

10.56

0 1

10.57

15 4

10.58

8 15

10.59

1+log3x
3+log2x+log2y

10.60

1+log9x
3+log3x+log3y

10.61

log431 logx3

10.62

log252 1logy

10.63

4log75 100·logx

10.64

7log23 20·logx

10.65

log25+4log2x+2log2y

10.66

log37+5log3x+3log3y

10.67

15(4log4x123log4y2log4z)

10.68

13(2log3xlog35log3ylog3z)

10.69

log25xy

10.70

log36xy

10.71

log2x3(x1)2

10.72

logx2(x+1)2

10.73

3.402

10.74

2.379

10.75

x=6

10.76

x=4

10.77

x=4

10.78

x=8

10.79

x=3

10.80

x=8

10.81

x=log43log71.933

10.82

x=log98log82.205

10.83

x=ln9+24.197

10.84

x=ln520.805

10.85

r9.3%

10.86

r11.9%

10.87

There will be 62,500 bacteria.

10.88

There will be 47,700 bacteria.

10.89

There will be 6.44 mg left.

10.90

There will be 31.5 mg left.

Section 10.1 Exercises

1.

8x+23 8x+11
8x2+26x+15

3.

24x+1 24x19
24x214x5

5.

6x29x 18x29x
6x39x2

7.

2x2+3 4x24x+3
2x3x2+4x2

9.

245 104 53

11.

250 14 77

13.

Function; not one-to-one

15.

One-to-one function

17.

Not a function Function; not one-to-one

19.

One-to-one function
Function; not one-to-one

21.

Inverse function: {(1,2),(2,4),(3,6),(4,8)}. Domain: {1,2,3,4}. Range: {2,4,6,8}.

23.

Inverse function: {(−2,0),(3,1),(7,2),(12,3)}. Domain: {−2,3,7,12}. Range: {0,1,2,3}.

25.

Inverse function: {(−3,2),(−1,−1),(1,0),(3,1)}. Domain: {−3,1,1,3}. Range: {−2,−1,0,1}.

27.
This figure shows a series of line segments from (negative 3, negative 4) to (0, negative 3) then to (2, negative 1), and then to (4, 3).
29.
This figure shows a series of line segments from (negative 1, 4) to (2, 3) then to (3, 0), and then to (4, negative 4).
31.

g(f(x))=x, and f(g(x))=x, so they are inverses.

33.

g(f(x))=x, and f(g(x))=x, so they are inverses.

35.

g(f(x))=x, and f(g(x))=x, so they are inverses.

37.

g(f(x))=x, and f(g(x))=x, so they are inverses (for nonnegative x).

39.

f−1(x)=x+12

41.

f−1(x)=x9

43.

f−1(x)=6x

45.

f−1(x)=x+76

47.

f−1(x)=x5−2

49.

f−1(x)=x6

51.

f−1(x)=3x+4

53.

f−1(x)=1x2

55.

f−1(x)=x2+2, x0

57.

f−1(x)=x3+3

59.

f−1(x)=x4+59, x0

61.

f−1(x)=x55−3

63.

Answers will vary.

Section 10.2 Exercises

65.
This figure shows a curve that passes through (negative 1, 1 over 2) through (0, 1) to (1, 2).
67.
This figure shows a curve that passes through (negative 1, 1 over 6) through (0, 1) to (1, 6).
69.
This figure shows a curve that passes through (negative 1, 2 over 3) through (0, 1) to (1, 3 over 2).
71.
This figure shows a curve that passes through (negative 1, 2) through (0, 1) to (1, 1 over 2).
73.
This figure shows a curve that passes through (negative1, 6) through (0, 1) to (1, 1 over 6).
75.
This figure shows a curve that passes through (negative 1, 5 over 2) through (0, 1) to (1, 2 over 5).
77.
This figure shows two functions. The first function f of x equals 4 to the x power is marked in blue and corresponds to a curve that passes through the points (negative 1, 1 over 4), (0, 1) and (1, 4). The second function g of x equals 4 to the x minus 1 power is marked in red and passes through the points (0, 1 over 4), (1, 1) and (2, 4).
79.
This figure shows two functions. The first function f of x equals 2 to the x power is marked in blue and corresponds to a curve that passes through the points (negative 1, 1 over 2), (0, 1) and (1, 2). The second function g of x equals 2 to the x minus 2 power is marked in red and passes through the points (0, 1 over 4), (1, 1 over 2), and (2, 1).
81.
This figure shows two functions. The first function f of x equals 3 to the x power is marked in blue and corresponds to a curve that passes through the points (negative 1, 1 over 3), (0, 1), and (1, 3). The second function g of x equals 3 to the x power plus 2 is marked in red and passes through the points (negative 2, 1), (negative 1, 3), and (0, 5).
83.
This figure shows two functions. The first function f of x equals 2 to the x power is marked in blue and corresponds to a curve that passes through the points (negative 1, 1 over 2), (0, 1), and (1, 2). The second function g of x equals 2 to the x power plus 1 is marked in red and passes through the points (negative 1, 1), (0, 2), and (1, 4).
85.
This figure shows an exponential curve that passes through (negative 3, 1 over 3), (negative 2, 1), and (0, 9).
87.
This figure shows an exponential that passes through (negative 1, 7 over 2), (0, 4), and (1, 5).
89.
This figure shows an exponential that passes through (2, 4), (3, 2), and (4, 1).
91.
This figure shows an exponential that passes through (1, 1 plus 1 over e), (0, 2), and (1, e).
93.
This figure shows an exponential that passes through (negative 1, negative 1 over 2), (0, negative 1), and (1, 2).
95.

x=4

97.

x=−1

99.

x=−1,x=1

101.

x=1

103.

x=−1

105.

x=2

107.

x=−1,x=2

109.

111.

113.

115.

$7,387.28 $7,434.57 $7,459.12

117.

$36,945.28

119.

162 bacteria

121.

288,929,825

123.

Answers will vary.

125.

Answers will vary.

Section 10.3 Exercises

127.

log232=5

129.

log5125=3

131.

log1100=−2

133.

logx36=13

135.

log17517=x

137.

log13181=4

139.

log4164=−3

141.

lnx=3

143.

64=26

145.

32=x5

147.

1=70

149.

9=91

151.

1,000=103

153.

43=ex

155.

x=11

157.

x=4

159.

x=125

161.

x=1243

163.

x=2

165.

x=−2

167.

2

169.

0

171.

13

173.

−2

175.

−3

177.

−2

179.
This figure shows the logarithmic curve going through the points (1 over 4, negative 1), (1, 0), and (4, 1).
181.
This figure shows that the logarithmic curve going through the points (1 over 7, negative 1), (1, 0), and (7, 1).
183.
This figure shows the logarithmic curve going through the points (2 over 5, negative 1), (1, 0), and (2.5, 1).
185.
This figure shows the logarithmic curve going through the points (1 over 5, 1), (1, 0), and (5, negative 1).
187.
This figure shows the logarithmic curve going through the points (3 over 5, 1), (1, 0), and (5 over 3, negative 1).
189.

a=9

191.

a=3

193.

a=233

195.

x=e4

197.

x=5

199.

x=17

201.

x=6

203.

x=3

205.

x=−55,x=55

207.

x=−5,x=5

209.

A whisper has a decibel level of 20 dB.

211.

The sound of a garbage disposal has a decibel level of 100 dB.

213.

The intensity of the 1994 Northridge earthquake in the Los Angeles area was about 40 times the intensity of the 2014 earthquake.

215.

Answers will vary.

217.

Answers will vary.

Section 10.4 Exercises

219.

0 1

221.

10 10

223.

15 −4

225.

3 −1

227.

3 7

229.

log58+log5y

231.

4+log3x+log3y

233.

3+logy

235.

log651

237.

3log5x

239.

4logy

241.

4ln16

243.

5log2x

245.

−3logx

247.

13log5x

249.

34lnx

251.

log23+5log2x+3log2y

253.

14log521+3log5y

255.

log54+log5a+3log5b
+4log5c2log5d

257.

23log3x34log3y

259.

12log3(3x+2y2)log352log3z

261.

13(log53+2log5xlog54
3log5ylog5z)

263.

2

265.

2

267.

log25x1

269.

log52xy

271.

log3x6y9

273.

0

275.

lnx3y4z2

277.

log(2x+3)2·x+1

279.

2.379

281.

1.674

283.

5.542

285.

Answers will vary.

287.

Answers will vary.

Section 10.5 Exercises

289.

x=7

291.

x=4

293.

x=1, x=3

295.

x=8

297.

x=3

299.

x=20

301.

x=3

303.

x=6

305.

x=53

307.

x=log74log26.209

309.

x=log112log43.404

311.

x=ln82.079

313.

x=log8log131.893

315.

x=ln320.901

317.

x=ln1630.924

319.

x=ln61.792

321.

x=ln8+13.079

323.

x=5

325.

x=−4,x=5

327.

a=3

329.

x=e9

331.

x=7

333.

x=3

335.

x=2

337.

x=6

339.

x=5

341.

x=log10log123.322

343.

x=ln753.054

345.

6.9%

347.

13.9 years

349.

122,070 bacteria

351.

8 times as large as the original population

353.

0.03 ml

355.

Answers will vary.

Review Exercises

357.

4x2+12x 16x2+12x 4x3+12x2

359.

−123 356 41

361.

Function; not one-to-one

363.

Function; not one-to-one Not a function

365.

Inverse function: {(10,−3),(5,−2),(2,−1),(1,0)}. Domain: {1,2,5,10}. Range: {−3,−2,−1,0}.

367.

g(f(x))=x, and f(g(x))=x, so they are inverses.

369.

f−1(x)=x+116

371.

f−1(x)=1x5

373.
This figure shows an exponential line passing through the points (negative 1, 1 over 4), (0, 1), and (1, 4).
375.
This figure shows an exponential line passing through the points (negative 1, 4 over 3), (0, 1), and (1, 3 over 4).
377.
This figure shows an exponential line passing through the points (negative 1, negative 59 over 23), (0, negative 2), and (1, negative7 over 10).
379.
This figure shows an exponential line passing through the points (negative 1, negative 1 over e), (0, negative 1), and (1, negative e).
381.

x=−2,x=2

383.

x=−1

385.

x=−3,x=5

387.

$163,323.40

389.

330,259,000

391.

log11,000=−3

393.

ln16=y

395.

100000=105

397.

x=5

399.

x=4

401.

0

403.
This figure shows a logarithmic line passing through the points (1 over 5, negative 1), (1, 0), and (5, 1).
405.
This figure shows a logarithmic line passing through the points (4 over 5, 1), (1, 0), and (5 over 4, negative 1).
407.

x=e−3

409.

x=8

411.

90 dB

413.

13 −9

415.

8 5

417.

4+logm

419.

5ln2

421.

17log4z

423.

log58+2log5a+6log5b
+log5c3log5d

425.

13(log67+2log6x13log6y
5log6z)

427.

log3x3y7

429.

log4y(y3)2

431.

5.047

433.

x=4

435.

x=3

437.

x=log101log26.658

439.

x=log7log131.771

441.

x=ln15+46.708

443.

11.6 years

445.

12.7 months

Practice Test

447.

48x17 48x+5
48x210x3

449.

Not a function One-to-one function

451.

f−1(x)=5x+9

453.

x=5

455.

$31,250.74 $31,302.29 $31,328.32

457.

343=73

459.

0

461. This figure shows a logarithmic line passing through (1 over 3, 1), (1, 0), and (3, 1).
463.

40 dB

465.

2+log5a+log5b

467.

14(log25+3log2x42log2y
7log2z)

469.

log6x(x+5)3

471.

x=6

473.

x=ln8+46.079

475.

1,921 bacteria


13.1.10: Chapter 10 is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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