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4.6e: Exercises - Exponential and Logarithmic Equations

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A: Concepts

Exercise 4.6e.1 

1) How can an exponential equation be solved?

2) When does an extraneous solution occur? How can an extraneous solution be recognized?

3) When can the one-to-one property of logarithms be used to solve an equation? When can it not be used?

Answer 1

Determine first if the equation can be rewritten so that each side uses the same base. If so, the exponents can be set equal to each other. If the equation cannot be rewritten so that each side uses the same base, then apply the logarithm to each side and use properties of logarithms to solve.

Answer 3

The one-to-one property can be used if both sides of the equation can be rewritten as a single logarithm with the same base. If so, the arguments can be set equal to each other, and the resulting equation can be solved algebraically. The one-to-one property cannot be used when each side of the equation cannot be rewritten as a single logarithm with the same base.

B: Solve Exponential Equations Using the 1-1 Property (like Bases)

Exercise 4.6e.2 

 For the following exercises, use like bases to solve the exponential equation.

4. 2x=16

5. 3x=81

6. 3x+4=27

7. 5x1=25

8. 23x+7=8

9. 25x2=16

10. 643x2=2

11. 812x+1=3

12. 815x32=0

13. 923x27=0

14. 4x2164=0

15. 16x22=0

16. 4x(2x+5)=64

17. 9x(x+1)=81

18. e3(3x21)e=0

19. 100x2107x3=0

20. 43v2=4v

21. 6443x=16

22. 32x+13x=243

23. 23n14=2n+2

24. 62553x+3=125

25. 363b362b=2162b

26. (164)3n8=26

Answers to odd exercises:

5. 4

7. 3

9. 65

11. 716

13. 16

15. ±12

17. 2,1

19. 12,3

21. x=13

23. n=1

25. b=65

C: Solve exponential equations using logarithms

Exercise 4.6e.3 

 For the following exercises, use logarithms to solve.

27. 9x10=1

28. 2e6x=13

29. er+1010=42

30. 2109a=29

31. 810p+77=24

32. 7e3n5+5=89

33. e3k+6=44

34. 5e9x88=62

35. 6e9x+8+2=74

36. 2x+1=52x1

37. e2xex132=0

38. 7e8x+85=95

39. 10e8x+3+2=8

40. 4e3x+37=53

41. 8e5x24=90

42. 32x+1=7x2

43. e2xex6=0

44. 3e33x+6=31

44.1 42x+3=5x2

44.2 9x3x=4x1

Answers to odd exercises:

27. x=10

29. No solution

31. p=log(178)7

33. k=ln(38)3

35. x=ln(383)89

37. x=ln12

39. x=ln(35)38

41. No solution

43. x=ln3

 Solve. Give the exact answer and the approximate answer rounded to the nearest thousandth.

45. 3x=5

46. 7x=2

47. 4x=9

48. 2x=10

49. 5x3=13

50. 3x+5=17

51. 72x+5=2

52. 35x9=11

53. 54x+3+6=4

54. 107x12=1

55. e2x35=0

56. e5x+110=0

57. 63x+13=7

58. 8109x+2=9

59. 15e3x=2

60. 7+e4x+1=10

61. 79ex=4

62. 36ex=0

63. 5x2=2

64. 32x2x=1

65. 100e27x=50

66. 6e12x=2

67. 31+ex=1

68. 21+3ex=1

Answers to odd exercises:

45. log5log31.465

47. log3log21.585

49. 3log5+log13log54.594

51. log25log72log72.322

53.

55. 3+ln522.305

57. 1log63log60.095

59. ln1330.855

61. ln31.099

63. ±log2log5±0.656

65. ln2270.026

67. ln20.693

 Find the x- and y-intercepts of the given function.

69. f(x)=3x+14

70. f(x)=23x11

71. f(x)=10x+1+2

72. f(x)=104x5

73. f(x)=ex2+1

74. f(x)=ex+44

 Use a u-substitution to solve the following.

75. 32x3x6=0 Hint: Let u=3x

76. 4x+2x20=0

77. 100x+10x12=0

78. 102x10x30=0

79. e2x3ex+2=0

80. e2x8ex+15=0

Answers to odd exercises:

69. x-intercept: (2log2log3log3,0);
y-intercept: (0,1)

71. No x-int.;  y-int.: (0,12)

73. No x-int.;  y-int.: (0,1+e2e2)

75. 1

77. log3

79. 0,ln2

D: Mixed exponential equations

Exercise 4.6e.4 

Solve each exponential equation. Find the exact answer and then approximate it to three decimal places.

81. 2x=74

82. 5x=110

83. 4x=112

84. 3x=89

85. ex=8

 

86. ex=16

87. (13)x=8

88. (12)x=6

89. 3ex+2=9

90. 6e2x=24

91. 2e3x=32

92. 4ex+1=16

93. 13ex=2

94. 14ex=3

95. ex1+4=12

96. ex+1+2=16

97. 64x17=216

98. 33x+1=81

99. ex2ex=e20

100. ex2e14=e5x

101. (12)x=10

102. 6x=91

103. 8ex+5=56

104. 7ex3=35

Answers to odd exercises:

81. x=log74log26.209

83. x=log112log43.404

85. x=ln82.079

87. x=log8log131.893

89. x=ln320.901

91. x=ln1630.924

93. x=ln61.792

95. x=ln8+13.079

97. x=5

99. x=4,x=5

101. x=log10log123.322

103. x=ln753.054

 For the following exercises, solve the exponential equation exactly. 

105. e3x15=0

106. 5x=125

107. 4x+132=0

108. 8x=4

109. 10x=7.21

110. 3x/14=110

111. 73x2=11

112. 423x20=0

 For the following exercises, solve each equation. Write the exact solution, and then approximate the answer to 3 decimal places.

113. e5x=17 

114. 1000(1.03)t=5000

115. 34x5=38 

116. 3(1.04)3t=8 

117. 7e3x5+7.9=47

118. 50e0.12t=10

119. log(0.7x9)=1+5log(5)

120. ln(3)+ln(4.4x+6.8)=2

Answers to odd exercises:

105. ln153

107. 32

109. log7.21

111. 23+log113log7

113. ln(17)50.567

115. x=log(38+5log(3))4log(3)2.078

117. x=13(5+ln(39.1/7))2.240

119. x=31340/.744655.714

E: Solve log equations by rewriting in exponential form

Exercise 4.6e.5 

 For the following exercises, use the definition of a logarithm to rewrite the equation as an exponential equation.

121. log(1100)=2 122. log324(18)=12

 For the following exercises, use the definition of a logarithm to solve the equation.

123. 5log7n=10

124. 8log9x=16

125. 4+log2(9k)=2

126. 2log(8n+4)+6=10

127. 104ln(98x)=6

 

 Find the x- and y-intercepts of the given function.

128.f(x)=log(x2)+1

129. f(x)=log(x+3)1

130. f(x)=log3(x+4)3

131. f(x)=log2(3x)4

132. f(x)=ln(x+1)+2

133. f(x)=ln(2x+5)6

Answers to odd exercises:

121. 102=1100

123. n=49

125. k=136

127. x=9e8

129. x-intercept: (7,0);
y-intercept: (0,log31)

131. x-intercept: (163,0);
y-intercept: None

133. x-intercept: (e652,0);
y-intercept: (0,ln56)

 Solve.

134. log3(2x+1)=2

135. log2(3x7)=5

136. log4(3x+5)=12

137. log(2x+20)=1

138. log(x2+3x+10)=1

139. log3x2=2

140. log5(x2+20)2=0

141. ln(x21)=0

142. log2(x+5)+log2(x+1)=5

143. log2(x5)+log2(x9)=5

144. log6x+log6(2x1)=2

145. log4x+log4(x6)=2

146. log2(x+1)log2(x2)=4

147. log3(2x+5)log3(x1)=2

148. ln(2x+1)lnx=2

149. lnxln(x1)=1

150. 2log2x=3+log2(x2)

151. 2log3x=2+log3(2x9)

152. log2(x+3)+log2(x+1)1=0

153. log2(x2)=2log2x

154. log2(x+2)+log2(1x)=1+log2(x+1) 155. logxlog(x+1)=1
Answers to odd exercises:

135. 13

137. 5

139. ±3

141. ±2

143. 13

145. 8

147. 2

149. ee1

151. 9

153. 1±5

155. Ø

F: Solve log equations using the 1-1 property

Exercise 4.6e.6 

 For the following exercises, use the one-to-one property of logarithms to solve.

156. ln(103x)=ln(4x)

157. log13(5n2)=log13(85n)

158. log(x+3)log(x)=log(74)

159. ln(3x)=ln(x26x)

160. log4(6m)=log4(3m)

161. ln(x2)ln(x)=ln(54)

162. log9(2n214n)=log9(45+n2)

163. ln(x210)+ln(9)=ln(10)

164. log(x+12)=log(x)+log(12)

165. ln(x)+ln(x3)=ln(7x)

166. log2(7x+6)=3

167. ln(7)+ln(24x2)=ln(14)

168. log8(x+6)log8(x)=log8(58)

169. ln(3)ln(33x)=ln(4)

170. log3(3x)log3(6)=log3(77)

171. log5(2x+4)=log5(3x6)

172. log4(7x)=log4(5x+14)

173. log2(x2)log2(6x5)=0

174. ln(2x1)=ln(3x)

175. log(x+5)log(2x+7)=0

176. ln(x2+4x)=2ln(x+1)

177. log32+2log3x=log3(7x3)

178. 2logxlog36=0

179. ln(x+3)+ln(x+1)=ln8

180. log5(x2)+log5(x5)=log510

Answers to odd exercises:

157. n=1

159. No solution

161. No solution

163. x=±103

165. x=10

167. x=0

169. x=34

171. 10

173. No solution

175. 2

177. 12,3

179. 1

G: Mixed log equations

Exercise 4.6e.7 

Solve for x. Give exact answer (not a decimal approximation).

182. log464=2log4x

183. log49=2logx

184. 3log3x=log327

185. 3log6x=log664

186. log5(4x2)=log510

187. log3(x2+3)=log34x

188. log3x+log3x=2

189. log4x+log4x=3

190. log2x+log2(x3)=2

191. log3x+log3(x+6)=3

192. logx+log(x+3)=1

193. logx+log(x15)=2

194. log(x+4)log(5x+12)=logx

195. log(x1)log(x+3)=log1x

196. log5(x+3)+log5(x6)=log510

197. log5(x+1)+log5(x5)=log57

198. log3(2x1)=log3(x+3)+log33

199. log(5x+1)=log(x+3)+log2

Answers to odd exercises:

183. x=7

185. x=4

187. x=1,x=3

189. x=8

191. x=3

193. x=20

195. x=3

197. x=6

199. x=53

Solve for x.

200. loga64=2

201. loga81=4

202. lnx=8

203. lnx=9

204. log5(3x8)=2

205. log4(7x+15)=3

206. lne5x=30

207. lne6x=18

208. 3logx=log125

209. 7log3x=log3128

210. log3x=0

211. log5x=2

212. log4(x+5)=0

213. log(2x7)=0

214. log3(x)+3=2

215. log9(x)5=4

216. ln(x5)=1

217. ln(3x)=2

218. lnx+3=2

219. ln(4x10)6=5

Answers to odd exercises:

201. a=3

203. x=e9

205. x=7

207. x=3

209. x=2

211: x=125

213: x=4

215. x=9

 

217. x=e232.5

219. x=e+1043.2

 Solve for x.

220. log6x+log6(x5)=log624

221. log9x+log9(x4)=log912

222. log4(x+2)log4(x1)=0

223. log6(x+9)+log6x=2

224. 7+log3(4x)=6

225. lnx+ln(x2)=ln4

226. log(42x)=log(4x)

227. log(4)+log(5x)=2

228. ln(2x+9)=ln(5x)

229. log11(2x27x)=log11(x2)

230. log(x2+13)=log(7x+3)

231. log9(3x)=log9(4x8)

232. ln(x)ln(x+3)=ln(6)

233. 3log2(10)log(x9)=log(44)

 

234. log2(x+2)log2(2x+9)=log2x  235. log6(x+1)log6(4x+10)=log61x
Answers to odd exercises:

221. x=6

223. x=3

225. 1+5

227. x=5

229. No solution

231. x=1152.2

233. x=101119.2

235. x=5

H: Inverses of Log and Exponent Functions

Exercise 4.6e.8 

 Find the inverse of the following functions.

237. f(x)=log2(x+5)

238. f(x)=4+log3x

239. f(x)=log(x+2)3

240. f(x)=ln(x4)+1

241. f(x)=ln(9x2)+5

242. f(x)=log6(2x+7)1

243. g(x)=e3x

244. g(x)=102x

245. g(x)=2x+3

246. g(x)=32x+5

247. g(x)=10x+43

248. g(x)=e2x1+1

Answers to odd exercises:

237. f1(x)=2x5

239. f1(x)=10x+32

241. f1(x)=ex5+29

243. g1(x)=lnx3

245. g1(x)=log2x3

247. g1(x)=log(x+3)4

I: Mixed log and exponential equations

Exercise 4.6e.9 

 Solve.

249. log(9x+5)=1+log(x5)

250. 2+log2(x2+1)=log213

251. e5x2e3x=0

252. 3x211=70

253. 23x5=0

254. log7(x+1)+log7(x1)=1

255. ln(4x1)1=lnx

256. log(20x+1)=logx+2

257. 31+e2x=2

258. 2e3x=4

259. 2e3x=e4x+1

260. 2logx+logx1=0

261. 3logx=log(x2)+2logx

262. 2ln3+lnx2=ln(x2+1)

Answers to odd exercises:

249. 55

251. 1

253. log253

255. 14e

257. ln(1/2)2

259. ln21

261.

 

J: Applications

Exercise 4.6e.10 

263. In chemistry, pH is a measure of acidity and is given by the formula pH=log(H+), where H+ is the hydrogen ion concentration (measured in moles of hydrogen per liter of solution.) Determine the hydrogen ion concentration if the pH of a solution is 4.

264. The volume of sound, L in decibels (dB), is given by the formula L=10log(I/1012) where I represents the intensity of the sound in watts per square meter. Determine the intensity of an alarm that emits 120 dB of sound.

265. An account with an initial deposit of $6,500 earns 7.25% annual interest, compounded continuously. How much will the account be worth after 20 years?

266. The formula for measuring sound intensity in decibels D is defined by the equation D=10log(II0), where I is the intensity of the sound in watts per square meter and I_0=10^{-12} is the lowest level of sound that the average person can hear. How many decibels are emitted from a jet plane with a sound intensity of 8\cdot 3\cdot 10^2 watts per square meter?

267. The population of a small town is modeled by the equation P=1650e^{0.5t} where t is measured in years. In approximately how many years will the town’s population reach 20,000?

268. Atmospheric pressure P in pounds per square inch is represented by the formula P=14.7e^{-0.21x}, where x is the number of miles above sea level. To the nearest foot, how high is the peak of a mountain with an atmospheric pressure of 8.369 pounds per square inch? (Hint: there are 5280 feet in a mile)

269. The magnitude M of an earthquake is represented by the equation M=\dfrac{2}{3}\log \left ( \dfrac{E}{E_0} \right ) where E is the amount of energy released by the earthquake in joules E_0=10^{4.4} and is the assigned minimal measure released by an earthquake. To the nearest hundredth, what would the magnitude be of an earthquake releasing 1.4\cdot 10^{13} joules of energy?

270. Use the definition of a logarithm along with the one-to-one property of logarithms to prove that b^{\log_b x}=x.

271. Recall the formula for continually compounding interest, y=Ae^{kt}.Use the definition of a logarithm along with properties of logarithms to solve the formula for time t such that t is equal to a single logarithm.

272. Recall the compound interest formula A=a\left ( 1+\frac{r}{k} \right )^{kt}.Use the definition of a logarithm along with properties of logarithms to solve the formula for time t

273. Newton’s Law of Cooling states that the temperature T of an object at any time t can be described by the equation T=T_s+(T_0-T_s)e^{-kt}, where T_s is the temperature of the surrounding environment, T_0 is the initial temperature of the object, and k is the cooling rate. Use the definition of a logarithm along with properties of logarithms to solve the formula for time t such that t is equal to a single logarithm.

Answers to odd exercises:

263. 10^{-4} moles per liter.

265. about \$27,710.24

267. about 5 years

269. about 5.83

271. t=\ln \left ( \left ( \dfrac{y}{A} \right )^{\frac{1}{k}} \right )

273. t=\ln \left ( \left ( \frac{T-T_s}{T_0-T_s} \right )^{-\frac{1}{k}} \right )

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4.6e: Exercises - Exponential and Logarithmic Equations is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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