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. 2−x=16 5. 3x=81 6. 3x+4=27 7. 5x−1=25 8. 23x+7=8 9. 25x−2=16 |
10. 643x−2=√2 11. 812x+1=√3 12. 81−5x−32=0 13. 92−3x−27=0 14. 4x2−1−64=0 15. 16x2−2=0 |
16. 4x(2x+5)=64 17. 9x(x+1)=81 18. e3(3x2−1)−e=0 19. 100x2−107x−3=0 20. 4−3v−2=4−v 21. 64⋅43x=16 |
22. 32x+1⋅3x=243 23. 2−3n⋅14=2n+2 24. 625⋅53x+3=125 25. 363b362b=2162−b 26. (164)3n⋅8=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. 9x−10=1 28. 2e6x=13 29. er+10−10=−42 30. 2⋅109a=29 31. −8⋅10p+7−7=−24 |
32. 7e3n−5+5=−89 33. e−3k+6=44 34. −5e9x−8−8=−62 35. −6e9x+8+2=−74 36. 2x+1=52x−1 |
37. e2x−ex−132=0 38. 7e8x+8−5=−95 39. 10e8x+3+2=8 40. 4e3x+3−7=53 41. 8e−5x−2−4=−90 |
42. 32x+1=7x−2 43. e2x−ex−6=0 44. 3e3−3x+6=−31 44.1 42x+3=5x−2 44.2 9x⋅3x=4x−1 |
- 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. 5x−3=13 50. 3x+5=17 51. 72x+5=2 |
52. 35x−9=11 53. 54x+3+6=4 54. 107x−1−2=1 55. e2x−3−5=0 56. e5x+1−10=0 57. 63x+1−3=7 |
58. 8−109x+2=9 59. 15−e3x=2 60. 7+e4x+1=10 61. 7−9e−x=4 62. 3−6e−x=0 63. 5x2=2 |
64. 32x2−x=1 65. 100e27x=50 66. 6e12x=2 67. 31+e−x=1 68. 21+3e−x=1 |
- Answers to odd exercises:
-
45. log5log3≈1.465
47. log3log2≈1.585
49. 3log5+log13log5≈4.594
51. log2−5log72log7≈−2.322
53. ∅
55. 3+ln52≈2.305
57. 1−log63log6≈0.095
59. ln133≈0.855
61. ln3≈1.099
63. ±√log2log5≈±0.656
65. −ln227≈−0.026
67. −ln2≈−0.693
★ Find the x- and y-intercepts of the given function.
69. f(x)=3x+1−4 70. f(x)=23x−1−1 |
71. f(x)=10x+1+2 72. f(x)=104x−5 |
73. f(x)=ex−2+1 74. f(x)=ex+4−4 |
★ Use a u-substitution to solve the following.
75. 32x−3x−6=0 Hint: Let u=3x 76. 4x+2x−20=0 |
77. 100x+10x−12=0 78. 102x−10x−30=0 |
79. e2x−3ex+2=0 80. e2x−8ex+15=0 |
- Answers to odd exercises:
-
69. x-intercept: (2log2−log3log3,0);
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. ex−1+4=12 |
96. ex+1+2=16 97. 64x−17=216 98. 33x+1=81 99. ex2ex=e20 |
100. ex2e14=e5x 101. (12)x=10 102. 6x=91 103. 8ex+5=56 104. 7ex−3=35 |
- Answers to odd exercises:
-
81. x=log74log2≈6.209
83. x=log112log4≈3.404
85. x=ln8≈2.079
87. x=log8log13≈−1.893
89. x=ln3−2≈−0.901
91. x=ln163≈0.924
93. x=ln6≈1.792
95. x=ln8+1≈3.079
97. x=5
99. x=−4,x=5
101. x=log10log12≈−3.322
103. x=ln7−5≈−3.054
★ For the following exercises, solve the exponential equation exactly.
105. e3x−15=0 106. 5x=125 |
107. 4x+1−32=0 108. 8x=4 |
109. 10x=7.21 110. 3x/14=110 |
111. 73x−2=11 112. 4⋅23x−20=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. 34x−5=38 |
116. 3(1.04)3t=8 117. 7e3x−5+7.9=47 118. 50e−0.12t=10 |
119. log(−0.7x−9)=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)5≈0.567
115. x=log(38+5log(3))4log(3)≈2.078
117. x=13(5+ln(39.1/7))≈2.240
119. x=−31340/.7≈−44655.714
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. 10−4ln(9−8x)=6
|
★ Find the x- and y-intercepts of the given function.
128.f(x)=log(x−2)+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. 10−2=1100
123. n=49
125. k=136
127. x=9−e8
129. x-intercept: (7,0);
y-intercept: (0,log3−1)131. x-intercept: (163,0);
y-intercept: None133. x-intercept: (e6−52,0);
y-intercept: (0,ln5−6)
★ Solve.
134. log3(2x+1)=2 135. log2(3x−7)=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(x2−1)=0 142. log2(x+5)+log2(x+1)=5 143. log2(x−5)+log2(x−9)=5 144. log6x+log6(2x−1)=2 145. log4x+log4(x−6)=2 146. log2(x+1)−log2(x−2)=4 147. log3(2x+5)−log3(x−1)=2 |
148. ln(2x+1)−lnx=2 149. lnx−ln(x−1)=1 150. 2log2x=3+log2(x−2) 151. 2log3x=2+log3(2x−9) 152. log2(x+3)+log2(x+1)−1=0 153. log2(x−2)=2−log2x |
154. log2(x+2)+log2(1−x)=1+log2(x+1) | 155. logx−log(x+1)=1 |
- Answers to odd exercises:
-
135. 13
137. −5
139. ±3
141. ±√2
143. 13
145. 8
147. 2
149. ee−1
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(10−3x)=ln(−4x) 157. log13(5n−2)=log13(8−5n) 158. log(x+3)−log(x)=log(74) 159. ln(−3x)=ln(x2−6x) 160. log4(6−m)=log4(3m) 161. ln(x−2)−ln(x)=ln(54) 162. log9(2n2−14n)=log9(−45+n2) 163. ln(x2−10)+ln(9)=ln(10) 164. log(x+12)=log(x)+log(12) 165. ln(x)+ln(x−3)=ln(7x) 166. log2(7x+6)=3 167. ln(7)+ln(2−4x2)=ln(14) 168. log8(x+6)−log8(x)=log8(58) |
169. ln(3)−ln(3−3x)=ln(4) 170. log3(3x)−log3(6)=log3(77) 171. log5(2x+4)=log5(3x−6) 172. log4(7x)=log4(5x+14) 173. log2(x−2)−log2(6x−5)=0 174. ln(2x−1)=ln(3x) 175. log(x+5)−log(2x+7)=0 176. ln(x2+4x)=2ln(x+1) 177. log32+2log3x=log3(7x−3) 178. 2logx−log36=0 179. ln(x+3)+ln(x+1)=ln8 180. log5(x−2)+log5(x−5)=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(4x−2)=log510 187. log3(x2+3)=log34x 188. log3x+log3x=2 189. log4x+log4x=3 190. log2x+log2(x−3)=2 |
191. log3x+log3(x+6)=3 192. logx+log(x+3)=1 193. logx+log(x−15)=2 194. log(x+4)−log(5x+12)=−logx 195. log(x−1)−log(x+3)=log1x 196. log5(x+3)+log5(x−6)=log510 197. log5(x+1)+log5(x−5)=log57 198. log3(2x−1)=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(3x−8)=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(2x−7)=0 214. log3(x)+3=2 |
215. log9(x)−5=−4 216. ln(x−5)=1 217. ln(3x)=2 218. ln√x+3=2 219. ln(4x−10)−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=e23≈2.5
219. x=e+104≈3.2
★ Solve for x.
220. log6x+log6(x−5)=log624 221. log9x+log9(x−4)=log912 222. log4(x+2)−log4(x−1)=0 223. log6(x+9)+log6x=2 224. −7+log3(4−x)=−6 |
225. lnx+ln(x−2)=ln4 226. log(4−2x)=log(−4x) 227. log(4)+log(−5x)=2 228. ln(2x+9)=ln(−5x) 229. log11(−2x2−7x)=log11(x−2) |
230. log(x2+13)=log(7x+3) 231. log9(3−x)=log9(4x−8) 232. ln(x)−ln(x+3)=ln(6) 233. 3log2(10)−log(x−9)=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=115≈2.2
233. x=10111≈9.2
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(x−4)+1 |
241. f(x)=ln(9x−2)+5 242. f(x)=log6(2x+7)−1 243. g(x)=e3x 244. g(x)=10−2x |
245. g(x)=2x+3 246. g(x)=32x+5 247. g(x)=10x+4−3 248. g(x)=e2x−1+1 |
- Answers to odd exercises:
-
237. f−1(x)=2x−5
239. f−1(x)=10x+3−2
241. f−1(x)=ex−5+29
243. g−1(x)=lnx3
245. g−1(x)=log2x−3
247. g−1(x)=log(x+3)−4
I: Mixed log and exponential equations
Exercise 4.6e.9
★ Solve.
249. log(9x+5)=1+log(x−5) 250. 2+log2(x2+1)=log213 251. e5x−2−e3x=0 252. 3x2−11=70 253. 23x−5=0 |
254. log7(x+1)+log7(x−1)=1 255. ln(4x−1)−1=lnx 256. log(20x+1)=logx+2 257. 31+e2x=2 258. 2e−3x=4 |
259. 2e3x=e4x+1 260. 2logx+logx−1=0 261. 3logx=log(x−2)+2logx 262. 2ln3+lnx2=ln(x2+1) |
- Answers to odd exercises:
-
249. 55
251. 1
253. log253
255. 14−e
257. ln(1/2)2
259. ln2−1
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/10−12) 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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