4.6e: Exercises - Exponential and Logarithmic Equations

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A: Concepts

Exercise $\PageIndex{1}$

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 $\PageIndex{2}$

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

4. $2^{-x}=16$

5. $3^{x}=81$

6. $3^{x+4}=27$

7. $5^{x-1}=25$

8. $2^{3 x+7}=8$

9. $2^{5 x-2}=16$

10. $64^{3 x-2}=\sqrt{2}$

11. $81^{2 x+1}=\sqrt{3}$

12. $8^{1-5 x}-32=0$

13. $9^{2-3 x}-27=0$

14. $4^{x^{2}-1}-64=0$

15. $16^{x^{2}}-2=0$

16. $4^{x(2 x+5)}=64$

17. $9^{x(x+1)}=81$

18. $e^{3\left(3 x^{2}-1\right)}-e=0$

19. $100^{x^{2}}-10^{7 x-3}=0$

20. $4^{-3v-2}=4^{-v}$

21. $64\cdot 4^{3x}=16$

22. $3^{2x+1}\cdot 3^x=243$

23. $2^{-3n}\cdot \frac{1}{4}=2^{n+2}$

24. $625\cdot 5^{3x+3}=125$

25. $\dfrac{36^{3b}}{36^{2b}}=216^{2-b}$

26. $\left (\frac{1}{64} \right )^{3n}\cdot 8=2^6$

Answers to odd exercises:

5. $4$

7. $3$

9. $\frac{6}{5}$

11. $-\frac{7}{16}$

13. $\frac{1}{6}$

15. $\pm \frac{1}{2}$

17. $-2,1$

19. $\frac{1}{2}, 3$

21. $x=-\frac{1}{3}$

23. $n=-1$

25. $b=\frac{6}{5}$

C: Solve exponential equations using logarithms

Exercise $\PageIndex{3}$

$ \bigstar $ For the following exercises, use logarithms to solve.

27. $9^{x-10}=1$

28. $2e^{6x}=13$

29. $e^{r+10}-10=-42$

30. $2\cdot 10^{9a}=29$

31. $-8\cdot 10^{p+7}-7=-24$

32. $7e^{3n-5}+5=-89$

33. $e^{-3k}+6=44$

34. $-5e^{9x-8}-8=-62$

35. $-6e^{9x+8}+2=-74$

36. $2^{x+1}=5^{2x-1}$

37. $e^{2x}-e^{x}-132=0$

38. $7e^{8x+8}-5=-95$

39. $10e^{8x+3}+2=8$

40. $4e^{3x+3}-7=53$

41. $8e^{-5x-2}-4=-90$

42. $3^{2x+1}=7^{x-2}$

43. $e^{2x}-e^{x}-6=0$

44. $3e^{3-3x}+6=-31$

44.1 $4^{2x+3}=5^{x-2}$

44.2 $9^x \cdot 3^x=4^{x-1}$

Answers to odd exercises:

27. $x=10$

29. No solution

31. $p=\log \left (\frac{17}{8} \right )-7$

33. $k=-\dfrac{\ln(38)}{3}$

35. $x=\dfrac{\ln \left( \frac{38}{3} \right) -8}{9}$

37. $x=\ln 12$

39. $x=\dfrac{\ln \left( \frac{3}{5} \right) -3}{8}$

41. No solution

43. $x=\ln 3$

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

45. $3^{x}=5$

46. $7^{x}=2$

47. $4^{x}=9$

48. $2^{x}=10$

49. $5^{x-3}=13$

50. $3^{x+5}=17$

51. $7^{2 x+5}=2$

52. $3^{5 x-9}=11$

53. $5^{4 x+3}+6=4$

54. $10^{7 x-1}-2=1$

55. $e^{2 x-3}-5=0$

56. $e^{5 x+1}-10=0$

57. $6^{3 x+1}-3=7$

58. $8-10^{9 x+2}=9$

59. $15-e^{3 x}=2$

60. $7+e^{4 x+1}=10$

61. $7-9 e^{-x}=4$

62. $3-6 e^{-x}=0$

63. $5^{x^{2}}=2$

64. $3^{2 x^{2}-x}=1$

65. $100 e^{27 x}=50$

66. $6 e^{12 x}=2$

67. $\dfrac{3}{1+e^{-x}}=1$

68. $\dfrac{2}{1+3 e^{-x}}=1$

Answers to odd exercises:

45. $\frac{\log 5}{\log 3} \approx 1.465$

47. $\frac{\log 3}{\log 2} \approx 1.585$

49. $\frac{3 \log 5+\log 13}{\log 5} \approx 4.594$

51. $\frac{\log 2-5 \log 7}{2 \log 7} \approx-2.322$

53. $\varnothing$

55. $\frac{3+\ln 5}{2} \approx 2.305$

57. $\frac{1-\log 6}{3 \log 6} \approx 0.095$

59. $\frac{\ln 13}{3} \approx 0.855$

61. $\ln 3 \approx 1.099$

63. $\pm \sqrt{\frac{\log 2}{\log 5}} \approx \pm 0.656$

65. $-\frac{\ln 2}{27} \approx-0.026$

67. $-\ln 2 \approx-0.693$

$ \bigstar $ Find the $x$- and $y$-intercepts of the given function.

69. $f(x)=3^{x+1}-4$

70. $f(x)=2^{3 x-1}-1$

71. $f(x)=10^{x+1}+2$

72. $f(x)=10^{4 x}-5$

73. $f(x)=e^{x-2}+1$

74. $f(x)=e^{x+4}-4$

$ \bigstar $ Use a $u$-substitution to solve the following.

75. $3^{2 x}-3^{x}-6=0$ Hint: Let $u=3^{x}$

76. $4^{ x}+2^{x}-20=0$

77. $100^{ x}+10^{x}-12=0$

78. $10^{2 x}-10^{x}-30=0$

79. $e^{2 x}-3 e^{x}+2=0$

80. $e^{2 x}-8 e^{x}+15=0$

Answers to odd exercises:

69. $x$-intercept: $\left(\dfrac{2 \log 2-\log 3}{\log 3}, 0\right)$;
$\quad$ $y$-intercept: $(0, −1)$

71. No $x$-int.; $\quad$ $y$-int.: $(0, 12)$

73. No $x$-int.; $\quad$ $y$-int.: $\left(0, \dfrac{1+e^{2}}{e^{2}}\right)$

75. $1$

77. $\log 3$

79. $0, \ln 2$

D: Mixed exponential equations

Exercise $\PageIndex{4}$

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

81. $2^{x}=74$

82. $5^{x}=110$

83. $4^{x}=112$

84. $3^{x}=89$

85. $e^{x}=8$

86. $e^{x}=16$

87. $\left(\frac{1}{3}\right)^{x}=8$

88. $\left(\frac{1}{2}\right)^{x}=6$

89. $3 e^{x+2}=9$

90. $6 e^{2 x}=24$

91. $2 e^{3 x}=32$

92. $4 e^{x+1}=16$

93. $\frac{1}{3} e^{x}=2$

94. $\frac{1}{4} e^{x}=3$

95. $e^{x-1}+4=12$

96. $e^{x+1}+2=16$

97. $6^{4 x-17}=216$

98. $3^{3 x+1}=81$

99. $\dfrac{e^{x^{2}}}{e^{x}}=e^{20}$

100. $\dfrac{e^{x^{2}}}{e^{14}}=e^{5 x}$

101. $\left(\frac{1}{2}\right)^{x}=10$

102. $6^{x}=91$

103. $8 e^{x+5}=56$

104. $7 e^{x-3}=35$

Answers to odd exercises:

81. $x=\frac{\log 74}{\log 2} \approx 6.209$

83. $x=\frac{\log 112}{\log 4} \approx 3.404$

85. $x=\ln 8 \approx 2.079$

87. $x=\dfrac{\log 8}{\log \frac{1}{3}} \approx-1.893$

89. $x=\ln 3-2 \approx-0.901$

91. $x=\frac{\ln 16}{3} \approx 0.924$

93. $x=\ln 6 \approx 1.792$

95. $x=\ln 8+1 \approx 3.079$

97. $x=5$

99. $x=-4, x=5$

101. $x=\dfrac{\log 10}{\log \frac{1}{2}} \approx-3.322$

103. $x=\ln 7-5 \approx-3.054$

$ \bigstar $ For the following exercises, solve the exponential equation exactly.

105. $e^{3x}−15=0$

106. $5^x=125$

107. $4^{x+1}−32=0$

108. $8^x=4$

109. $10^x=7.21$

110. $3^{x/14}=\frac{1}{10}$

111. $7^{3x−2}=11$

112. $4⋅2^{3x}−20=0$

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

113. $e^{5x}=17$

114. $1000(1.03)^t=5000$

115. $3^{4x-5}=38$

116. $3(1.04)^{3t}=8$

117. $7e^{3x-5}+7.9=47$

118. $50e^{-0.12t}=10$

119. $\log(-0.7x-9)=1+5\log(5)$

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

Answers to odd exercises:

105. $\dfrac{ln15}{3}$

107. $\frac{3}{2}$

109. $\log7.21$

111. $\frac{2}{3}+\dfrac{log11}{3log7}$

113. $\dfrac{\ln (17)}{5}
\approx 0.567$

115. $x=\dfrac{\log (38+5\log (3))}{4\log(3)}
\approx 2.078$

117. $ x=\frac{1}{3} (5+\ln(39.1/7))
\approx 2.240$

119. $x= - 31340/.7
\approx -44655.714$

E: Solve log equations by rewriting in exponential form

Exercise $\PageIndex{5}$

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

121. $\log \left ( \frac{1}{100} \right )=-2$

122. $\log _{324}(18)=\dfrac{1}{2}$

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

123. $5\log _{7}n=10$

124. $-8\log _{9}x=16$

125. $4+\log _{2}(9k)=2$

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

127. $10-4\ln (9-8x)=6$

$ \bigstar $ 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. $f(x)=\log _{3}(x+4)-3$

131. $f(x)=\log _{2}(3 x)-4$

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

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

Answers to odd exercises:

121. $10^{-2}=\dfrac{1}{100}$

123. $n=49$

125. $k=\frac{1}{36}$

127. $x=\dfrac{9-e}{8}$

129. $x$-intercept: $(7,\; 0)$;
$\quad$ $y$-intercept: $(0,\; \log 3-1)$

131. $x$-intercept: $(\frac{16}{3},\; 0)$;
$\quad$ $y$-intercept: None

133. $x$-intercept: $\left(\dfrac{e^{6}-5}{2},\; 0\right)$;
$\quad$ $y$-intercept: $(0,\; \ln 5-6)$

$ \bigstar $ Solve.

134. $\log _{3}(2 x+1)=2$

135. $\log _{2}(3 x-7)=5$

136. $\log _{4}(3 x+5)=\frac{1}{2}$

137. $\log (2 x+20)=1$

138. $\log \left(x^{2}+3 x+10\right)=1$

139. $\log _{3} x^{2}=2$

140. $\log _{5}\left(x^{2}+20\right)-2=0$

141. $\ln \left(x^{2}-1\right)=0$

142. $\log _{2}(x+5)+\log _{2}(x+1)=5$

143. $\log _{2}(x-5)+\log _{2}(x-9)=5$

144. $\log _{6} x+\log _{6}(2 x-1)=2$

145. $\log _{4} x+\log _{4}(x-6)=2$

146. $\log _{2}(x+1)-\log _{2}(x-2)=4$

147. $\log _{3}(2 x+5)-\log _{3}(x-1)=2$

148. $\ln (2 x+1)-\ln x=2$

149. $\ln x-\ln (x-1)=1$

150. $2 \log _{2} x=3+\log _{2}(x-2)$

151. $2 \log _{3} x=2+\log _{3}(2 x-9)$

152. $\log _{2}(x+3)+\log _{2}(x+1)-1=0$

153. $\log _{2}(x-2)=2-\log _{2} x$

154. $\log _{2}(x+2)+\log _{2}(1-x)=1+\log _{2}(x+1)$

155. $\log x-\log (x+1)=1$

Answers to odd exercises:

135. $13$

137. $−5$

139. $±3$

141. $\pm \sqrt{2}$

143. $13$

145. $8$

147. $2$

149. $\frac{e}{e-1}$

151. $9$

153. 1$\pm \sqrt{5}$

155. $Ø$

F: Solve log equations using the 1-1 property

Exercise $\PageIndex{6}$

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

156. $\ln (10-3x)=\ln (-4x)$

157. $\log_{13} (5n-2)=\log_{13} (8-5n)$

158. $\log (x+3)-\log (x)=\log (74)$

159. $\ln (-3x)=\ln (x^2-6x)$

160. $\log_4 (6-m)=\log_4 (3m)$

161. $\ln (x-2)-\ln (x)=\ln (54)$

162. $\log_9 (2n^2-14n)=\log_9 (-45+n^2)$

163. $\ln (x^2-10)+\ln (9)=\ln (10)$

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

165. $\ln (x)+\ln (x-3)=\ln (7x)$

166. $\log_2 (7x+6)=3$

167. $\ln (7)+\ln (2-4x^2)=\ln (14)$

168. $\log_8 (x+6)-\log_8 (x)=\log_8 (58)$

169. $\ln (3)-\ln (3-3x)=\ln (4)$

170. $\log_3 (3x)-\log_3 (6)=\log_3 (77)$

171. $\log _{5}(2 x+4)=\log _{5}(3 x-6)$

172. $\log _{4}(7 x)=\log _{4}(5 x+14)$

173. $\log _{2}(x-2)-\log _{2}(6 x-5)=0$

174. $\ln (2 x-1)=\ln (3 x)$

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

176. $\ln \left(x^{2}+4 x\right)=2 \ln (x+1)$

177. $\log _{3} 2+2 \log _{3} x=\log _{3}(7 x-3)$

178. $2 \log x-\log 36=0$

179. $\ln (x+3)+\ln (x+1)=\ln 8$

180. $\log _{5}(x-2)+\log _{5}(x-5)=\log _{5} 10$

Answers to odd exercises:

157. $n=1$

159. No solution

161. No solution

163. $x=\pm \frac{10}{3}$

165. $x=10$

167. $x=0$

169. $x=\frac{3}{4}$

171. $10$

173. No solution

175. $−2$

177. $\frac{1}{2} ,\; 3$

179. $1$

G: Mixed log equations

Exercise $\PageIndex{7}$

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

182. $\log _{4} 64=2 \log _{4} x$

183. $\log 49=2 \log x$

184. $3 \log _{3} x=\log _{3} 27$

185. $3 \log _{6} x=\log _{6} 64$

186. $\log _{5}(4 x-2)=\log _{5} 10$

187. $\log _{3}\left(x^{2}+3\right)=\log _{3} 4 x$

188. $\log _{3} x+\log _{3} x=2$

189. $\log _{4} x+\log _{4} x=3$

190. $\log _{2} x+\log _{2}(x-3)=2$

191. $\log _{3} x+\log _{3}(x+6)=3$

192. $\log x+\log (x+3)=1$

193. $\log x+\log (x-15)=2$

194. $\log (x+4)-\log (5 x+12)=-\log x$

195. $\log (x-1)-\log (x+3)=\log \frac{1}{x}$

196. $\log _{5}(x+3)+\log _{5}(x-6)=\log _{5} 10$

197. $\log _{5}(x+1)+\log _{5}(x-5)=\log _{5} 7$

198. $\log _{3}(2 x-1)=\log _{3}(x+3)+\log _{3} 3$

199. $\log (5 x+1)=\log (x+3)+\log 2$

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. $x=\frac{5}{3}$

$ \bigstar $ Solve for $x$.

200. $\log _{a} 64=2$

201. $\log _{a} 81=4$

202. $\ln x=-8$

203. $\ln x=9$

204. $\log _{5}(3 x-8)=2$

205. $\log _{4}(7 x+15)=3$

206. $\ln e^{5 x}=30$

207. $\ln e^{6 x}=18$

208. $3 \log x=\log 125$

209. $7 \log _{3} x=\log _{3} 128$

210. $\log_3x=0$

211. $\log_5x=−2$

212. $\log_4(x+5)=0$

213. $\log(2x−7)=0$

214. $\log_3 (x)+3=2$

215. $\log_9 (x)-5=-4$

216. $\ln (x-5)=1$

217. $\ln (3x)=2$

218. $\ln\sqrt{x+3}=2$

219. $\ln (4x-10)-6=-5$

Answers to odd exercises:

201. $a=3$

203. $x=e^{9}$

205. $x=7$

207. $x=3$

209. $x=2$

211: $x=\frac{1}{25}$

213: $x=4$

215. $x=9$

217. $x=\frac{e^2}{3}\approx 2.5$

219. $x=\frac{e+10}{4}\approx 3.2$

$ \bigstar $ Solve for $x$.

220. $\log _{6} x+\log _{6}(x-5)=\log _{6} 24$

221. $\log _{9} x+\log _{9}(x-4)=\log _{9} 12$

222. $\log_4(x+2)−\log_4(x−1)=0$

223. $\log_6(x+9)+\log_6x=2$

224. $-7+\log_3 (4-x)=-6$

225. $\ln x+\ln(x−2)=\ln4$

226. $\log (4-2x)=\log (-4x)$

227. $\log (4)+\log (-5x)=2$

228. $\ln (2x+9)=\ln (-5x)$

229. $\log_{11} (-2x^2 -7x)=\log_{11} (x-2)$

230. $\log (x^2+13)=\log (7x+3)$

231. $\log_9 (3-x)=\log_9 (4x-8)$

232. $\ln (x)-\ln (x+3)=\ln (6)$

233. $\dfrac{3}{\log _2(10)}-\log (x-9)=\log (44)$

234. $\log _{2}(x+2)-\log _{2}(2 x+9)=-\log _{2} x$

235. $\log _{6}(x+1)-\log _{6}(4 x+10)=\log _{6} \frac{1}{x}$

Answers to odd exercises:

221. $x=6$

223. $x=3$

225. $1+\sqrt{5}$

227. $x=-5$

229. No solution

231. $x=\frac{11}{5}\approx 2.2$

233. $x=\frac{101}{11}\approx 9.2$

235. $x=5$

H: Inverses of Log and Exponent Functions

Exercise $\PageIndex{8}$

$ \bigstar $ Find the inverse of the following functions.

237. $f(x)=\log _{2}(x+5)$

238. $f(x)=4+\log _{3} x$

239. $f(x)=\log (x+2)-3$

240. $f(x)=\ln (x-4)+1$

241. $f(x)=\ln (9 x-2)+5$

242. $f(x)=\log _{6}(2 x+7)-1$

243. $g(x)=e^{3 x}$

244. $g(x)=10^{-2 x}$

245. $g(x)=2^{x+3}$

246. $g(x)=3^{2 x}+5$

247. $g(x)=10^{x+4}-3$

248. $g(x)=e^{2 x-1}+1$

Answers to odd exercises:

237. $f^{-1}(x)=2^{x}-5$

239. $f^{-1}(x)=10^{x+3}-2$

241. $f^{-1}(x)=\frac{e^{x-5}+2}{9}$

243. $g^{-1}(x)=\frac{\ln x}{3}$

245. $g^{-1}(x)=\log _{2} x-3$

247. $g^{-1}(x)=\log (x+3)-4$

I: Mixed log and exponential equations

Exercise $\PageIndex{9}$

$ \bigstar $ Solve.

249. $\log (9 x+5)=1+\log (x-5)$

250. $2+\log _{2}\left(x^{2}+1\right)=\log _{2} 13$

251. $e^{5 x-2}-e^{3 x}=0$

252. $3^{x^{2}}-11=70$

253. $2^{3 x}-5=0$

254. $\log _{7}(x+1)+\log _{7}(x-1)=1$

255. $\ln (4 x-1)-1=\ln x$

256. $\log (20 x+1)=\log x+2$

257. $\frac{3}{1+e^{2 x}}=2$

258. $2 e^{-3 x}=4$

259. $2 e^{3 x}=e^{4 x+1}$

260. $2 \log x+\log x-1=0$

261. $3 \log x=\log (x-2)+2 \log x$

262. $2 \ln 3+\ln x^{2}=\ln \left(x^{2}+1\right)$

Answers to odd exercises:

249. $55$

251. $1$

253. $\frac{\log _{2} 5}{3}$

255. $\frac{1}{4-e}$

257. $\frac{\ln (1 / 2)}{2}$

259. $\ln 2-1$

261. $\emptyset$

J: Applications

Exercise $\PageIndex{10}$

263. In chemistry, pH is a measure of acidity and is given by the formula $\mathrm{pH}=-\log \left(H^{+}\right)$, 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 $L=10 \log \left(I / 10^{-12}\right)$ 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 $D=10\log \left ( \frac{I}{I_0} \right )$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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