Intermediate Algebra

# Chapter 10

Intermediate AlgebraChapter 10

### Try It

10.1

$15x+115x+1$ $15x−915x−9$
$15x2−7x−215x2−7x−2$

10.2

$24x−2324x−23$ $24x−2324x−23$
$24x2−38x+1524x2−38x+15$

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: ${−4,−3,−2,−1}.{−4,−3,−2,−1}.$

10.11 10.12 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

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

10.19 10.20 10.21 10.22 10.23 10.24 10.25 10.26 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

$22,332.9622,332.96$
$22,362.4922,362.49$ $22,377.3722,377.37$

10.32

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

10.33

She will find 166 bacteria.

10.34

She 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

$27=3327=33$ $1=x01=x0$
$110=10−1110=10−1$

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 $1313$ $−2−2$

10.43 10.44 10.45 10.46 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

$log43−1log43−1$ $logx−3logx−3$

10.62

$log25−2log25−2$ $1−logy1−logy$

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 5,870,061 bacteria.

10.89

There will be 6.43 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.

$24x+124x+1$ $24x−1924x−19$
$24x2+14x−524x2+14x−5$

5.

$6x2−9x6x2−9x$ $18x2−9x18x2−9x$
$6x3−9x26x3−9x2$

7.

$2x2+32x2+3$ $4x2−4x+34x2−4x+3$
$2x3−x2+4x−22x3−x2+4x−2$

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. 29. 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.

$f−1(x)=x2+2f−1(x)=x2+2$, $x≥0x≥0$

57.

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

59.

$f−1(x)=x4+59f−1(x)=x4+59$, $x≥0x≥0$

61.

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

63.

### Section 10.2 Exercises

65. 67. 69. 71. 73. 75. 77. 79. 81. 83. 85. 87. 89. 91. 93. 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.

$7,387.287,387.28$ $7,434.577,434.57$ $7,459.127,459.12$

117.

$36,945.28 36,945.28$

119.

159 bacteria

121.

288,929,825

123.

125.

### 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. 181. 183. 185. 187. 189.

$a = 9 a = 9$

191.

$a = 3 a = 3$

193.

$a = 24 3 a = 24 3$

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.

217.

### 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.

287.

### 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.

### 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. 375. 377. 379. 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. 405. 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.

$48x−1748x−17$ $48x+548x+5$
$48x2−10x−348x2−10x−3$

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.

$31,250.7431,250.74$ $31,302.2931,302.29$ $31,328.3231,328.32$

457.

$343 = 7 3 343 = 7 3$

459.

0

461. 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.

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