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

$f−1(x)=x+35f−1(x)=x+35$

10.16

$f−1(x)=x−58f−1(x)=x−58$

10.17

$f−1(x)=x5+23f−1(x)=x5+23$

10.18

$f−1(x)=x4+76f−1(x)=x4+76$

10.19 10.20 10.21 10.22 10.23 10.24 10.25 10.26 10.27

$x=2x=2$

10.28

$x=4x=4$

10.29

$x=−1,x=2x=−1,x=2$

10.30

$x=−2,x=3x=−2,x=3$

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

$log25+4log2x+2log2ylog25+4log2x+2log2y$

10.66

$log37+5log3x+3log3ylog37+5log3x+3log3y$

10.67

$15(4log4x−12−3log4y−2log4z)15(4log4x−12−3log4y−2log4z)$

10.68

$13(2log3x−log35−log3y−log3z)13(2log3x−log35−log3y−log3z)$

10.69

$log25xylog25xy$

10.70

$log36xylog36xy$

10.71

$log2x3(x−1)2log2x3(x−1)2$

10.72

$logx2(x+1)2logx2(x+1)2$

10.73

$3.4023.402$

10.74

$2.3792.379$

10.75

$x=6x=6$

10.76

$x=4x=4$

10.77

$x=4x=4$

10.78

$x=8x=8$

10.79

$x=3x=3$

10.80

$x=8x=8$

10.81

$x=log43log7≈1.933x=log43log7≈1.933$

10.82

$x=log98log8≈2.205x=log98log8≈2.205$

10.83

$x=ln9+2≈4.197x=ln9+2≈4.197$

10.84

$x=ln52≈0.805x=ln52≈0.805$

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.

$f−1(x)=x+12f−1(x)=x+12$

41.

$f−1(x)=x9f−1(x)=x9$

43.

$f−1(x)=6xf−1(x)=6x$

45.

$f−1(x)=x+76f−1(x)=x+76$

47.

$f−1(x)=x−5−2f−1(x)=x−5−2$

49.

$f−1(x)=x−6f−1(x)=x−6$

51.

$f−1(x)=x+43f−1(x)=x+43$

53.

$f−1(x)=1x−2f−1(x)=1x−2$

55.

$f−1(x)=x2+2f−1(x)=x2+2$, $x≥0x≥0$

57.

$f−1(x)=x3+3f−1(x)=x3+3$

59.

$f−1(x)=x4+59f−1(x)=x4+59$, $x≥0x≥0$

61.

$f−1(x)=x5−5−3f−1(x)=x5−5−3$

63.

### Section 10.2 Exercises

65. 67. 69. 71. 73. 75. 77. 79. 81. 83. 85. 87. 89. 91. 93. 95.

$x=4x=4$

97.

$x=−1x=−1$

99.

$x=−1,x=1x=−1,x=1$

101.

$x=1x=1$

103.

$x=−1x=−1$

105.

$x=2x=2$

107.

$x=−1,x=2x=−1,x=2$

109.

111.

113.

115.

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

117.

$36,945.2836,945.28$

119.

159 bacteria

121.

288,929,825

123.

125.

### Section 10.3 Exercises

127.

$log232=5log232=5$

129.

$log5125=3log5125=3$

131.

$log1100=−2log1100=−2$

133.

$logx63=13logx63=13$

135.

$log17175=xlog17175=x$

137.

$log13181=4log13181=4$

139.

$log4164=−3log4164=−3$

141.

$lnx=3lnx=3$

143.

$64=2664=26$

145.

$32=x532=x5$

147.

$1=701=70$

149.

$9=919=91$

151.

$1,000=1031,000=103$

153.

$43=ex43=ex$

155.

$x=11x=11$

157.

$x=4x=4$

159.

$x=125x=125$

161.

$x=1243x=1243$

163.

$x=2x=2$

165.

$x=−2x=−2$

167.

2

169.

0

171.

$1313$

173.

$−2−2$

175.

$−3−3$

177.

$−2−2$

179. 181. 183. 185. 187. 189.

$a=9a=9$

191.

$a=3a=3$

193.

$a=243a=243$

195.

$x=e4x=e4$

197.

$x=5x=5$

199.

$x=17x=17$

201.

$x=6x=6$

203.

$x=3x=3$

205.

$x=−55,x=55x=−55,x=55$

207.

$x=−5,x=5x=−5,x=5$

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.

$log58+log5ylog58+log5y$

231.

$4+log3x+log3y4+log3x+log3y$

233.

$3+logy3+logy$

235.

$log65−1log65−1$

237.

$3−log5x3−log5x$

239.

$4−logy4−logy$

241.

$4−ln164−ln16$

243.

$5log2x5log2x$

245.

$−3logx−3logx$

247.

$13log5x13log5x$

249.

$43lnx43lnx$

251.

$log23+5log2x+3log2ylog23+5log2x+3log2y$

253.

$14log521+3log5y14log521+3log5y$

255.

$log54+log5a+3log5blog54+log5a+3log5b$
$+4log5c−2log5d+4log5c−2log5d$

257.

$23log3x−3−4log3y23log3x−3−4log3y$

259.

$12log3(3x+2y2)−log35−2log3z12log3(3x+2y2)−log35−2log3z$

261.

$13(log53+2log5x−log5413(log53+2log5x−log54$
$−3log5y−log5z)−3log5y−log5z)$

263.

2

265.

2

267.

$log25x−1log25x−1$

269.

$log52xylog52xy$

271.

$log3x6y9log3x6y9$

273.

0

275.

$lnx3y4z2lnx3y4z2$

277.

$log(2x+3)2·x+1log(2x+3)2·x+1$

279.

$2.3792.379$

281.

$1.6741.674$

283.

$5.5425.542$

285.

287.

### Section 10.5 Exercises

289.

$x=7x=7$

291.

$x=4x=4$

293.

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

295.

$x=8x=8$

297.

$x=3x=3$

299.

$x=20x=20$

301.

$x=3x=3$

303.

$x=6x=6$

305.

$x=53x=53$

307.

$x=log74log2≈6.209x=log74log2≈6.209$

309.

$x=log112log4≈3.404x=log112log4≈3.404$

311.

$x=ln8≈2.079x=ln8≈2.079$

313.

$x=log8log13≈−1.893x=log8log13≈−1.893$

315.

$x=ln3−2≈−0.901x=ln3−2≈−0.901$

317.

$x=ln163≈0.924x=ln163≈0.924$

319.

$x=ln6≈1.792x=ln6≈1.792$

321.

$x=ln8+1≈3.079x=ln8+1≈3.079$

323.

$x=5x=5$

325.

$x=−4,x=5x=−4,x=5$

327.

$a=3a=3$

329.

$x=e9x=e9$

331.

$x=7x=7$

333.

$x=3x=3$

335.

$x=2x=2$

337.

$x=6x=6$

339.

$x=5x=5$

341.

$x=log10log12≈−3.322x=log10log12≈−3.322$

343.

$x=ln7−5≈−3.054x=ln7−5≈−3.054$

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.

$f−1(x)=x+116f−1(x)=x+116$

371.

$f−1(x)=1x−5f−1(x)=1x−5$

373. 375. 377. 379. 381.

$x=−2,x=2x=−2,x=2$

383.

$x=−1x=−1$

385.

$x=−3,x=5x=−3,x=5$

387.

$163,323.40163,323.40$

389.

330,259,000

391.

$log11,000=−3log11,000=−3$

393.

$ln16=yln16=y$

395.

$100000=105100000=105$

397.

$x=5x=5$

399.

$x=4x=4$

401.

0

403. 405. 407.

$x=e−3x=e−3$

409.

$x=8x=8$

411.

90 dB

413.

13 $−9−9$

415.

8 5

417.

$4+logm4+logm$

419.

$5−ln25−ln2$

421.

$17log4z17log4z$

423.

$log58+2log5a+6log5blog58+2log5a+6log5b$
$+log5c−3log5d+log5c−3log5d$

425.

$13(log67+2log6x−1−3log6y13(log67+2log6x−1−3log6y$
$−5log6z)−5log6z)$

427.

$log3x3y7log3x3y7$

429.

$logy4(y−3)2logy4(y−3)2$

431.

5.047

433.

$x=4x=4$

435.

$x=3x=3$

437.

$x=log101log2≈6.658x=log101log2≈6.658$

439.

$x=log7log13≈−1.771x=log7log13≈−1.771$

441.

$x=ln15+4≈6.708x=ln15+4≈6.708$

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.

$f−1(x)=x+95f−1(x)=x+95$

453.

$x=5x=5$

455.

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

457.

$343=73343=73$

459.

0

461. 463.

40 dB

465.

$2+log5a+log5b2+log5a+log5b$

467.

$14(log25+3log2x−4−2log2y14(log25+3log2x−4−2log2y$
$−7log2z)−7log2z)$

469.

$logx6(x+5)3logx6(x+5)3$

471.

$x=6x=6$

473.

$x=ln8+4≈6.079x=ln8+4≈6.079$

475.

1,921 bacteria