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Be Prepared

10.1

f4=5f4=5; gf4=32gf4=32

10.2

x = 2 3 y + 4 x = 2 3 y + 4

10.3

x x

10.4

x x

10.5

11; 11

10.6

1212; 33

10.7

x = 9 , x = 9 x = 9 , x = 9

10.8

1 9 1 9

10.9

x = 7 x = 7

10.10

11; aa

10.11

x 2 y 1 3 x 2 y 1 3

10.12

2.565 2.565

10.13

x = 4 , x = 4 x = 4 , x = 4

10.14

x = 2 , x = 3 x = 2 , x = 3

10.15

x = 5 , x = 1 x = 5 , x = 1

Try It

10.1

15x+115x+1 15x915x9
15x27x215x27x2

10.2

24x2324x23 24x2324x23
24x238x+1524x238x+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)}.{(4,0),(7,1),(10,2),(13,3)}. Domain: {4,7,10,13}.{4,7,10,13}. Range: {0,1,2,3}.{0,1,2,3}.

10.10

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

10.13

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

10.14

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

10.15

f −1 ( x ) = x + 3 5 f −1 ( x ) = x + 3 5

10.16

f −1 ( x ) = x 5 8 f −1 ( x ) = x 5 8

10.17

f −1 ( x ) = x 5 + 2 3 f −1 ( x ) = x 5 + 2 3

10.18

f −1 ( x ) = x 4 + 7 6 f −1 ( x ) = x 4 + 7 6

10.27

x = 2 x = 2

10.28

x = 4 x = 4

10.29

x = −1 , x = 2 x = −1 , x = 2

10.30

x = −2 , x = 3 x = −2 , x = 3

10.31

$22,332.96$22,332.96
$22,362.49$22,362.49 $22,377.37$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=2log39=2
log77=12log77=12 log13127=xlog13127=x

10.36

log464=3log464=3
log443=13log443=13 log12132=xlog12132=x

10.37

64=4364=43
1=x01=x0 1100=10−21100=10−2

10.38

27=3327=33 1=301=30
110=10−1110=10−1

10.39


x=8x=8 x=125x=125 x=2x=2

10.40



x=9x=9 x=243x=243 x=3x=3

10.41


2 1212 −5−5

10.42

2 1313 −2−2

10.47



a=11a=11
x=e7x=e7

10.48



a=4a=4
x=e9x=e9

10.49



x=13x=13
x=2x=2

10.50



x=6x=6
x=1x=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+log3x1+log3x
3+log2x+log2y3+log2x+log2y

10.60

1+log9x1+log9x
3+log3x+log3y3+log3x+log3y

10.61

log431log431 logx3logx3

10.62

log252log252 1logy1logy

10.63

4log754log75 100·logx100·logx

10.64

7log237log23 20·logx20·logx

10.65

log 2 5 + 4 log 2 x + 2 log 2 y log 2 5 + 4 log 2 x + 2 log 2 y

10.66

log 3 7 + 5 log 3 x + 3 log 3 y log 3 7 + 5 log 3 x + 3 log 3 y

10.67

1 5 ( 4 log 4 x 1 2 3 log 4 y 2 log 4 z ) 1 5 ( 4 log 4 x 1 2 3 log 4 y 2 log 4 z )

10.68

1 3 ( 2 log 3 x log 3 5 log 3 y log 3 z ) 1 3 ( 2 log 3 x log 3 5 log 3 y log 3 z )

10.69

log 2 5 x y log 2 5 x y

10.70

log 3 6 x y log 3 6 x y

10.71

log 2 x 3 ( x 1 ) 2 log 2 x 3 ( x 1 ) 2

10.72

log x 2 ( x + 1 ) 2 log x 2 ( x + 1 ) 2

10.73

3.402 3.402

10.74

2.379 2.379

10.75

x = 6 x = 6

10.76

x = 4 x = 4

10.77

x = 4 x = 4

10.78

x = 8 x = 8

10.79

x = 3 x = 3

10.80

x = 8 x = 8

10.81

x = log 43 log 7 1.933 x = log 43 log 7 1.933

10.82

x = log 98 log 8 2.205 x = log 98 log 8 2.205

10.83

x = ln 9 + 2 4.197 x = ln 9 + 2 4.197

10.84

x = ln 5 2 0.805 x = ln 5 2 0.805

10.85

r 9.3 % r 9.3 %

10.86

r 11.9 % r 11.9 %

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+238x+23 8x+118x+11
8x2+26x+158x2+26x+15

3.

24x+124x+1 24x1924x19
24x214x524x214x5

5.

6x29x6x29x 18x29x18x29x
6x39x26x39x2

7.

2x2+32x2+3 4x24x+34x24x+3
2x3x2+4x22x3x2+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)}.{(1,2),(2,4),(3,6),(4,8)}. Domain: {1,2,3,4}.{1,2,3,4}. Range: {2,4,6,8}.{2,4,6,8}.

23.

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

25.

Inverse function: {(−3,2),(−1,−1),(1,0),(3,1)}.{(−3,2),(−1,−1),(1,0),(3,1)}. Domain: {−3,1,1,3}.{−3,1,1,3}. Range: {−2,−1,0,1}.{−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,g(f(x))=x, and f(g(x))=x,f(g(x))=x, so they are inverses.

33.

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

35.

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

37.

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

39.

f −1 ( x ) = x + 12 f −1 ( x ) = x + 12

41.

f −1 ( x ) = x 9 f −1 ( x ) = x 9

43.

f −1 ( x ) = 6 x f −1 ( x ) = 6 x

45.

f −1 ( x ) = x + 7 6 f −1 ( x ) = x + 7 6

47.

f −1 ( x ) = x 5 −2 f −1 ( x ) = x 5 −2

49.

f −1 ( x ) = x 6 f −1 ( x ) = x 6

51.

f −1 ( x ) = x + 4 3 f −1 ( x ) = x + 4 3

53.

f −1 ( x ) = 1 x 2 f −1 ( x ) = 1 x 2

55.

f−1(x)=x2+2f−1(x)=x2+2, x0x0

57.

f −1 ( x ) = x 3 + 3 f −1 ( x ) = x 3 + 3

59.

f−1(x)=x4+59f−1(x)=x4+59, x0x0

61.

f −1 ( x ) = x 5 5 −3 f −1 ( x ) = x 5 5 −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 x = 4

97.

x = −1 x = −1

99.

x = −1 , x = 1 x = −1 , x = 1

101.

x = 1 x = 1

103.

x = −1 x = −1

105.

x = 2 x = 2

107.

x = −1 , x = 2 x = −1 , x = 2

109.

111.

113.

115.

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

117.

$ 36,945.28 $ 36,945.28

119.

162 bacteria

121.

288,929,825

123.

Answers will vary.

125.

Answers will vary.

Section 10.3 Exercises

127.

log 2 32 = 5 log 2 32 = 5

129.

log 5 125 = 3 log 5 125 = 3

131.

log 1 100 = −2 log 1 100 = −2

133.

log x 6 3 = 1 3 log x 6 3 = 1 3

135.

log 17 17 5 = x log 17 17 5 = x

137.

log 1 3 1 81 = 4 log 1 3 1 81 = 4

139.

log 4 1 64 = −3 log 4 1 64 = −3

141.

ln x = 3 ln x = 3

143.

64 = 2 6 64 = 2 6

145.

32 = x 5 32 = x 5

147.

1 = 7 0 1 = 7 0

149.

9 = 9 1 9 = 9 1

151.

1,000 = 10 3 1,000 = 10 3

153.

43 = e x 43 = e x

155.

x = 11 x = 11

157.

x = 4 x = 4

159.

x = 125 x = 125

161.

x = 1 243 x = 1 243

163.

x = 2 x = 2

165.

x = −2 x = −2

167.

2

169.

0

171.

1 3 1 3

173.

−2 −2

175.

−3 −3

177.

−2 −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 a = 9

191.

a = 3 a = 3

193.

a = 2 3 3 a = 2 3 3

195.

x = e 4 x = e 4

197.

x = 5 x = 5

199.

x = 17 x = 17

201.

x = 6 x = 6

203.

x = 3 x = 3

205.

x = −5 5 , x = 5 5 x = −5 5 , x = 5 5

207.

x = −5 , x = 5 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 −4−4

225.

33 −1−1

227.

3 7

229.

log 5 8 + log 5 y log 5 8 + log 5 y

231.

4 + log 3 x + log 3 y 4 + log 3 x + log 3 y

233.

3 + log y 3 + log y

235.

log 6 5 1 log 6 5 1

237.

3 log 5 x 3 log 5 x

239.

4 log y 4 log y

241.

4 ln 16 4 ln 16

243.

5 log 2 x 5 log 2 x

245.

−3 log x −3 log x

247.

1 3 log 5 x 1 3 log 5 x

249.

4 3 ln x 4 3 ln x

251.

log 2 3 + 5 log 2 x + 3 log 2 y log 2 3 + 5 log 2 x + 3 log 2 y

253.

1 4 log 5 21 + 3 log 5 y 1 4 log 5 21 + 3 log 5 y

255.

log 5 4 + log 5 a + 3 log 5 b log 5 4 + log 5 a + 3 log 5 b
+ 4 log 5 c 2 log 5 d + 4 log 5 c 2 log 5 d

257.

2 3 log 3 x 3 4 log 3 y 2 3 log 3 x 3 4 log 3 y

259.

1 2 log 3 ( 3 x + 2 y 2 ) log 3 5 2 log 3 z 1 2 log 3 ( 3 x + 2 y 2 ) log 3 5 2 log 3 z

261.

1 3 ( log 5 3 + 2 log 5 x log 5 4 1 3 ( log 5 3 + 2 log 5 x log 5 4
3 log 5 y log 5 z ) 3 log 5 y log 5 z )

263.

2

265.

2

267.

log 2 5 x 1 log 2 5 x 1

269.

log 5 2 x y log 5 2 x y

271.

log 3 x 6 y 9 log 3 x 6 y 9

273.

0

275.

ln x 3 y 4 z 2 ln x 3 y 4 z 2

277.

log ( 2 x + 3 ) 2 · x + 1 log ( 2 x + 3 ) 2 · x + 1

279.

2.379 2.379

281.

1.674 1.674

283.

5.542 5.542

285.

Answers will vary.

287.

Answers will vary.

Section 10.5 Exercises

289.

x = 7 x = 7

291.

x = 4 x = 4

293.

x=1,x=1, x=3x=3

295.

x = 8 x = 8

297.

x = 3 x = 3

299.

x = 20 x = 20

301.

x = 3 x = 3

303.

x = 6 x = 6

305.

x = 5 3 x = 5 3

307.

x = log 74 log 2 6.209 x = log 74 log 2 6.209

309.

x = log 112 log 4 3.404 x = log 112 log 4 3.404

311.

x = ln 8 2.079 x = ln 8 2.079

313.

x = log 8 log 1 3 1.893 x = log 8 log 1 3 1.893

315.

x = ln 3 2 0.901 x = ln 3 2 0.901

317.

x = ln 16 3 0.924 x = ln 16 3 0.924

319.

x = ln 6 1.792 x = ln 6 1.792

321.

x = ln 8 + 1 3.079 x = ln 8 + 1 3.079

323.

x = 5 x = 5

325.

x = −4 , x = 5 x = −4 , x = 5

327.

a = 3 a = 3

329.

x = e 9 x = e 9

331.

x = 7 x = 7

333.

x = 3 x = 3

335.

x = 2 x = 2

337.

x = 6 x = 6

339.

x = 5 x = 5

341.

x = log 10 log 1 2 3.322 x = log 10 log 1 2 3.322

343.

x = ln 7 5 3.054 x = ln 7 5 3.054

345.

6.9 % 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+12x4x2+12x 16x2+12x16x2+12x 4x3+12x24x3+12x2

359.

−123−123 356 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)}.{(10,−3),(5,−2),(2,−1),(1,0)}. Domain: {1,2,5,10}.{1,2,5,10}. Range: {−3,−2,−1,0}.{−3,−2,−1,0}.

367.

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

369.

f −1 ( x ) = x + 11 6 f −1 ( x ) = x + 11 6

371.

f −1 ( x ) = 1 x 5 f −1 ( x ) = 1 x 5

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 x = −2 , x = 2

383.

x = −1 x = −1

385.

x = −3 , x = 5 x = −3 , x = 5

387.

$ 163,323.40 $ 163,323.40

389.

330,259,000

391.

log 1 1,000 = −3 log 1 1,000 = −3

393.

ln 16 = y ln 16 = y

395.

100000 = 10 5 100000 = 10 5

397.

x = 5 x = 5

399.

x = 4 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 x = e −3

409.

x = 8 x = 8

411.

90 dB

413.

13 −9−9

415.

8 5

417.

4 + log m 4 + log m

419.

5 ln 2 5 ln 2

421.

1 7 log 4 z 1 7 log 4 z

423.

log 5 8 + 2 log 5 a + 6 log 5 b log 5 8 + 2 log 5 a + 6 log 5 b
+ log 5 c 3 log 5 d + log 5 c 3 log 5 d

425.

1 3 ( log 6 7 + 2 log 6 x 1 3 log 6 y 1 3 ( log 6 7 + 2 log 6 x 1 3 log 6 y
5 log 6 z ) 5 log 6 z )

427.

log 3 x 3 y 7 log 3 x 3 y 7

429.

log y 4 ( y 3 ) 2 log y 4 ( y 3 ) 2

431.

5.047

433.

x = 4 x = 4

435.

x = 3 x = 3

437.

x = log 101 log 2 6.658 x = log 101 log 2 6.658

439.

x = log 7 log 1 3 1.771 x = log 7 log 1 3 1.771

441.

x = ln 15 + 4 6.708 x = ln 15 + 4 6.708

443.

11.6 years

445.

12.7 months

Practice Test

447.

48x1748x17 48x+548x+5
48x210x348x210x3

449.

Not a function One-to-one function

451.

f −1 ( x ) = x + 9 5 f −1 ( x ) = x + 9 5

453.

x = 5 x = 5

455.

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

457.

343 = 7 3 343 = 7 3

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 + log 5 a + log 5 b 2 + log 5 a + log 5 b

467.

1 4 ( log 2 5 + 3 log 2 x 4 2 log 2 y 1 4 ( log 2 5 + 3 log 2 x 4 2 log 2 y
7 log 2 z ) 7 log 2 z )

469.

log x 6 ( x + 5 ) 3 log x 6 ( x + 5 ) 3

471.

x = 6 x = 6

473.

x = ln 8 + 4 6.079 x = ln 8 + 4 6.079

475.

1,921 bacteria

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