4.5e: Exercises - Properties of Logarithms

A: Concepts

Exercise \(\PageIndex{A}\) 

1. How does the power rule for logarithms help when solving logarithms with the form \(\log _b(\sqrt[n]{x})\)
2. Write the Product Property in your own words. Does it apply to each of the following? \(\log _{a} 5 x, \log _{a}(5+x)\). Why or why not?
3. Write the Power Property in your own words. Does it apply to each of the following? \(\log _{a} x^{p},\left(\log _{a} x\right)^{r}\). Why or why not?
4. Use an example to show that \(\log (a+b) \neq \log a+\log b ?\)
5. Explain how to find the value of \(\log _{7} 15\) using your calculator.
6. What does the change-of-base formula do? Why is it useful when using a calculator?

Answers to odd exercises:

1. Any root expression can be rewritten as an expression with a rational exponent so that the power rule can be applied, making the logarithm easier to calculate. Thus, \(\log _b \left ( x^{\frac{1}{n}} \right ) = \dfrac{1}{n}\log_{b}(x)\).

3. Answers may vary

5. Answers may vary

B: Basic simplification of logs

Exercise \(\PageIndex{B}\) 

\( \bigstar \) For the following exercises, use properties of logarithms to evaluate without using a calculator.

7. \(\log _{7} 1\) 8. \(\log _{1 / 2} 2\) 9. \(\log 10^{14}\) 10. \(\log 10^{-23}\) 11. \(\log _{3} 3^{10}\) 12. \(\log _{6} 6\) 13. \(\ln e^{7}\) 14. \(\ln \left(\frac{1}{e}\right)\)

15. \(\log _{1 / 2}\left(\frac{1}{2}\right)\) 16. \(\log _{1 / 5} 5\) 17. \(\log _{3 / 4}\left(\frac{4}{3}\right)\) 18. \(\log _{2 / 3} 1\) 19. \(2^{\log _{2} 100}\) 20. \(3^{\log _{3} 1}\) 21. \(10^{\log 18}\) 22. \(e^{\ln 23}\)

23. \(e^{\ln x^{2}}\) 24. \(e^{\ln e^{x}}\) 25. a. \(\log _{12} 1\), b. \(\ln e\) 26. a. \(\log _{4} 1\), b. \(\log _{8} 8\) 27. a. \(5^{\log _{5} 10}\), b. \(\log _{4} 4^{10}\) 28. a. \(3^{\log _{3} 6}\), b. \(\log _{2} 2^{7}\) 29. a. \(6^{\log _{6} 15}\), b. \(\log _{8} 8^{-4}\) 30. a. \(8^{\log _{8} 7}\), b. \(\log _{6} 6^{-2}\)

32. a. \(10^{\log \sqrt{5}}\), b. \(\log 10^{-2}\) 33. a. \(10^{\log \sqrt{3}}\), b. \(\log 10^{-1}\) 34. a. \(e^{\ln 4}\), b. \(\ln e^{2}\) 35. a. \(e^{\ln 3}\), b. \(\ln e^{7}\) 36. \(\log _3 \left ( \frac{1}{9} \right )-3\log _3 (3)\) 37. \(6\log _8 (2)+\frac{\log _8 (64)}{3\log _8 (4)}\) 38. \(2\log _9 (3)-4\log _9 (3) \\ +\log _9 \left (\frac{1}{729} \right )\)

\( \bigstar \) Find \(a\):

39. \(\ln a=1\) 40. \(\log a=-1\)

41. \(\log _{9} a=-1\) 42. \(\log _{12} a=1\)

43. \(\log _{2} a=5\) 44. \(\log a=13\)

45. \(2^{a}=7\) 46. \(e^{a}=23\)

47. \(\log _{a} 4^{5}=5\) 48. \(\log _{a} 10=1\)

Answers to odd exercises:

7. \(0\) 9. \(14\) 11. \(10\) 13. \(7\)

15. \(1\) 17. \(−1\) 19. \(100\) 21. \(18\)

23. \(x^{2}\) 25. \(0\), \(1\) 27. \(10\), \(10\) 29. \(15\), \(-4\)

33. \(\sqrt{3}\), \(-1\) 35. \(3\), \(7\) 37. \(3\)  39. \(e\)

41. \(\frac{1}{9}\) 43. \(2^{5}=32\) 45. \(\log _{2} 7\) 47. \(4\)

C: Expand logarithms

Exercise \(\PageIndex{C}\) 

\( \bigstar \) In the following exercises, use the Product Property of Logarithms to write each logarithm as a sum of logarithms. Simplify if possible.

49. \(\log _{5} 8 y\)

50. \(\log _{4} 6 x\)

51. \(\log _{3} 81 x y\)

52. \(\log _{2} 32 x y\)

53. \(\log 1000 y\)

54. \(\log 100 x\)

\( \bigstar \) In the following exercises, use the Quotient Property of Logarithms to write each logarithm as a sum of logarithms. Simplify if possible.

55. \(\log _{6} \dfrac{5}{6} \\[6pt] \) 56. \(\log _{3} \dfrac{3}{8}\)

57. \(\log _{5} \dfrac{125}{x} \\[6pt] \) 58. \(\log _{4} \dfrac{16}{y}\)

59. \(\log \dfrac{10,000}{y} \\[6pt] \) 60. \(\log \dfrac{x}{10}\)

61. \(\ln \dfrac{e^{4}}{16} \\[6pt] \) 62. \(\ln \dfrac{e^{3}}{3}\)

\( \bigstar \) In the following exercises, use the Power Property of Logarithms to expand each. Simplify if possible.

63. \(\log _{2} x^{5}\) 64. \(\log _{3} x^{2}\)

65. \(\log x^{-3}\) 66. \(\log x^{-2}\)

67. \(\log _{5} \sqrt[3]{x}\) 68. \(\log _{4} \sqrt{x}\)

69. \(\ln x^{\sqrt[3]{4}}\) 70. \(\ln x^{\sqrt{3}}\)

Answers to odd exercises:

49. \(\log _{5} 8+\log _{5} y\) 51. \(4+\log _{3} x+\log _{3} y\)

53. \(3+\log y\) 55. \(\log _{6} 5-1\)

57. \(3-\log _{5} x\) 59. \(4-\log y\)

61. \(4-\ln 16\) 63. \(5\log _{2} x\)

65. \(-3 \log x\) 67. \(\frac{1}{3} \log _{5} x\)

69. \(\sqrt[3]{4} \ln x\)

D: Expand logarithms

Exercise \(\PageIndex{D}\)  

\( \bigstar \) In the following exercises, use the Properties of Logarithms to expand the logarithm. Simplify if possible.

71. \(\log _{2}\left(3 x^{5} y^{3}\right) \\[6pt] \) 72. \(\log _{5}\left(4 x^{6} y^{4}\right) \\[6pt] \) 73. \(\log _{5}\left(\sqrt[4]{21} y^{3}\right) \\[6pt] \) 74. \(\log _{3}\left(\sqrt{2} x^{2}\right) \\[6pt] \) 75.\(\log _{4}(x y) \\[6pt] \) 76. \(\log (6 x)\)

77. \(\log _{3}\left(9 x^{2}\right) \\[6pt] \) 78. \(\log _{2}\left(32 x^{7}\right) \\[6pt] \) 79. \(\ln \left(3 y^{2}\right) \\[6pt] \) 80. \(\log \left(100 x^{2}\right) \\[6pt] \) 81. \(\log \left(10 x^{2} y^{3}\right) \\[6pt] \) 82. \(\log _{2}\left(2 x^{4} y^{5}\right) \\[6pt] \)

83. \(\log _{6}\left(36(x+y)^{4}\right) \\[6pt] \) 84. \(\ln \left[(e^{4}(x-y)^{3}\right) \\[6pt] \) 85. \(\log _{7}(2 \sqrt{x y}) \\[6pt] \) 86. \(\ln (2 x \sqrt{y}) \\[6pt] \) 87. \(\log _b (7x\cdot 2y) \\[6pt] \) 88. \(\ln (3ab\cdot 5c) \\[6pt] \)

89. \(\log \left ( \sqrt{x^3y^{-4}} \right ) \\[6pt] \) 90. \(\log _2 (y^x) \\[6pt] \) 91. \(\log \left ( x^2y^3 \sqrt[3]{x^2y^5} \right ) \\[6pt] \) 92. \(\log (x^4y) \\[6pt] \) 93. \(\log_5\sqrt{125xy^3} \\[6pt] \) 94. \(\ln (a\sqrt[3]{b}) \\[6pt] \)

Answers to odd exercises:

71. \(\log _{2} 3+5 \log _{2} x+3 \log _{2} y\) 73. \(\frac{1}{4} \log _{5} 21+3 \log _{5} y\) 75. \(\log _{4} x+\log _{4} y\) 77. \(2+2 \log _{3} x\)

79. \(\ln 3+2 \ln y\) 81. \(1+2 \log x+3 \log y\) 83. \(2+4 \log _{6}(x+y)\) 85. \(\log _{7} 2+\frac{1}{2} \log _{7} x+\frac{1}{2} \log _{7} y\)

87. \(\log _b 14 +\log _b (x)+\log _b (y)\) 89. \(\frac{3}{2}\log (x)-2\log (y)\) 91. \(\frac{8}{3}\log (x)+\frac{14}{3}\log (y)\) 93: \(\frac{3}{2}+\frac{1}{2}log_5x+\frac{3}{2}log_5y\)

\( \bigstar \) In the following exercises, use the Properties of Logarithms to expand the logarithm. Simplify if possible.

95. \(\log _{5} \dfrac{4 a b^{3} c^{4}}{d^{2}} \\[6pt] \) 96. \(\log _{3} \dfrac{x y^{2}}{z^{2}} \\[6pt] \) 97. \(\log _{2}\left(\dfrac{x}{y^{2}}\right) \\[6pt] \) 98. \(\log _{5}\left(\dfrac{25}{x}\right) \\[6pt] \)

99. \(\log _{3}\left(\dfrac{x^{3}}{y z^{2}}\right) \\[6pt] \) 100. \(\log \left(\dfrac{x}{y^{3} z^{2}}\right) \\[6pt] \) 101. \(\log _{5}\left(\dfrac{1}{x^{2} y z}\right) \\[6pt] \) 102. \(\log _{4}\left(\dfrac{1}{16 x^{2} z^{3}}\right) \\[6pt] \)

103. \(\log \left(\dfrac{100 x^{3}}{(y+10)^{3}}\right) \\[6pt] \) 104. \(\log \left(\dfrac{2(x+y)^{3}}{z^{2}}\right) \\[6pt] \) 105. \(\log_b \left ( \dfrac{13}{17} \right ) \\[6pt] \) 106. \(\log_4 \left ( \dfrac{\frac{x}{z}}{w} \right ) \\[6pt] \)

107. \(\ln \left ( \dfrac{1}{4e^k} \right ) \\[6pt] \) 108. \(ln \dfrac{9e^5}{4b} \\[6pt] \) 109. \(\log \left ( \dfrac{x^{15}y^{13}}{z^{19}} \right ) \\[6pt] \) 110. \(\ln \left ( \dfrac{a^{-2}}{b^{-4}c^{5}} \right ) \\[6pt] \)

Answers to odd exercises:

95. \(\log _{5} 4+\log _{5} a+3 \log _{5} b \\ +4 \log _{5} c-2 \log _{5} d\) 97. \(\log _{2} x-2 \log _{2} y\)

99. \(3 \log _{3} x-\log _{3} y-2 \log _{3} z\) 101. \(-2 \log _{5} x-\log _{5} y-\log _{5} z\) 103. \(2+3 \log x-3 \log (y+10)\)

105. \(\log _b (13)-\log _b (17)\) 107. \(-\ln(4)  - k\) 109. \(15\log (x)+13\log (y)-19\log (z)\)

 \( \bigstar \) In the following exercises, use the Properties of Logarithms to expand the logarithm. Simplify if possible.

111. \(\log _{3} \dfrac{\sqrt[3]{x^{2}}}{27 y^{4}} \\[6pt] \) 112. \(\log _{4} \dfrac{\sqrt{x}}{16 y^{4}} \\[6pt] \) 113. \(\log _{3} \dfrac{\sqrt{3 x+2 y^{2}}}{5 z^{2}} \\[6pt] \) 114. \(\log _{2} \dfrac{\sqrt{2 x+y^{2}}}{8 z^{2}} \\[6pt] \)

115. \(\log _{5} \sqrt[3]{\dfrac{3 x^{2}}{4 y^{3} z}} \\[6pt] \) 116. \(\log _{2} \sqrt[4]{\dfrac{5 x^{3}}{2 y^{2} z^{4}}} \\[6pt] \) 117. \(\log _{3}\left(\dfrac{x^{2} \sqrt[3]{y}}{z}\right) \\[6pt] \)

118. \(\log _{7}\left(\dfrac{x}{\sqrt[5]{(y+z)^{3}}}\right) \\[6pt] \) 119. \(\log _{5}\left(\dfrac{x^{3}}{\sqrt[3]{y z^{2}}}\right) \\[6pt] \) 120. \(\log \left(\dfrac{x^{2}}{\sqrt[5]{y^{3} z^{2}}}\right) \\[6pt] \)

122. \(\ln \left ( y\sqrt{\dfrac{y}{1-y}} \right ) \\[6pt] \) 123. \(\ln(\dfrac{6}{\sqrt{e^3}})\) 124. \(\log_4\dfrac{\sqrt[3]{xy}}{64} \\[6pt] \)

Answers to odd exercises:

111. \(\frac{2}{3} \log _{3} x-3-4 \log _{3} y \\[6pt] \) 113. \(\frac{1}{2} \log _{3}\left(3 x+2 y^{2}\right)-\log _{3} 5-2 \log _{3} z\)

115. \(\frac{1}{3} \left( \log _{5} 3+2 \log _{5} x-\log _{5} 4\right) \\ -3 \log _{5} y-\log _{5} z ) \) 117. \(2 \log _{3} x+\frac{1}{3} \log _{3} y-\log _{3} z\)

119. \(3 \log _{5} x-\frac{1}{3} \log _{5} y-\frac{2}{3} \log _{5} z \\[6pt] \) 123: \(−\frac{3}{2}+\ln6\)

E: Apply Log Properties

Exercise \(\PageIndex{E}\) 

\( \bigstar \) Given \(\log _{3} x=a, \log _{3} y=b\), and \(\log _{3} z=c\), write the following logarithms in terms of \(a, b\), and and \(c\).

125. \(\log _{3}\left(27 x^{2} y^{3} z\right)\)

126. \(\log _{3}\left(x y^{3} \sqrt{z}\right)\)

127. \(\log _{3}\left(\frac{9 x^{2} y}{z^{3}}\right)\)

128. \(\log _{3}\left(\frac{\sqrt[3]{x}}{y z^{2}}\right)\)

\( \bigstar \) Given \(\log _{b} 2=0.43, \log _{b} 3=0.68\), and \(\log _{b} 7=1.21\), calculate the following. (Hint: Expand using sums, differences, and quotients of the factors \(2, 3\), and \(7\).)

129. \(\log _{b} 42\)

130. \(\log _{b}(36)\)

131. \(\log _{b}\left(\frac{28}{9}\right)\)

132. \(\log _{b} \sqrt{21}\)

\( \bigstar \) Expand using the properties of the logarithm and then approximate using a calculator to the nearest tenth.

133. \(\log \left(3.10 \times 10^{25}\right)\)

134. \(\log \left(1.40 \times 10^{-33}\right)\)

135. \(\ln \left(6.2 e^{-15}\right)\)

136. \(\ln \left(1.4 e^{22}\right)\)

Answers to odd exercises:

125. \(3+2 a+3 b+c\) \(\;\;\) 127. \(2+2 a+b-3 c\) \(\;\;\) 129. \(2.32\)  \(\;\;\) 131. \(0.71\) \(\;\;\) 133. \(\log (3.1)+25  \approx 25.5 \)  \( \;\; \) 135. \(\ln (6.2)-15 \approx-13.2\)

F: Condense Logarithms

Exercise \(\PageIndex{F}\) 

\( \bigstar \) For the following exercises, condense each expression to a single logarithm with a coefficient \(1\) using the properties of logarithms.

137. \(\log x+\log y\) 138. \(\log _{3} x-\log _{3} y\) 139. \(\log_b(28)-\log_b(7)\) 140. \(\ln (a)-\ln (d)-\ln (c)\) 141. \(\ln (7)+\ln (x)+\ln (y)\) 142. \(\log_3 2+\log_3 a +\log_3 11+\log_3 b \) 143. \(\ln \left ( 6x^9 \right )-\ln \left (3x^2 \right )\) 144. \(\log \left ( 2x^4 \right )+\log \left (3x^5 \right )\)

145. \(\log 4+\log 25\) 146. \(\log _{6} 4+\log _{6} 9\) 147. \(\log _{3} 36-\log _{3} 4\) 148. \(\log _{2} 80-\log _{2} 5\) 149. \(\log _{2} 5-\log _{2}(x-1)\) 150. \(\log _{3} 4+\log _{3}(x+1)\) 151. \(\log _{5} 2-\log _{5} x-\log _{5} y\) 152. \(\log _{7} 3+\log _{7} x-\log _{7} y\)

153. \(\log (x+1)+\log (x-1)\) 154. \(\log _{2}(x+2)+\log _{2}(x+1)\) 155. \(\ln \left(x^{2}+2 x+1\right)-\ln (x+1)\) 156. \(\ln \left(x^{2}-9\right)-\ln (x+3)\) 157. \(\log _{5}\left(x^{3}-8\right)-\log _{5}(x-2)\) 158. \(\log _{3}\left(x^{3}+1\right)-\log _{3}(x+1)\) 159. \(\log x+\log (x+5)-\log \left(x^{2}-25\right)\) 160. \(\log (2 x+1)+\log (x-3) \\  -\log \left(2 x^{2}-5 x-3\right)\)

Answers to odd exercises:

137. \(\log (x y)\) 139. \(\log_b(4)\) 141. \(\ln(7xy)\)

143. \(\ln \left ( 2x^7 \right )\) 145. \(2\) 147. \(2\)

149. \(\log _{2} \frac{5}{x-1}\) 151. \(\log _{5} \frac{2}{x y}\) 153. \(\log \left(x^{2}-1\right)\)

155. \(\ln (x+1)\) 157. \(\log _{5}\left(x^{2}+2 x+4\right)\) 159. \(\log \left(\frac{x}{x-5}\right)\)

\( \bigstar \) For the following exercises, condense each expression to a single logarithm with a coefficient \(1\) using the properties of logarithms.

161. \(-\log_b\left ( \frac{1}{7} \right )\) 162. \(\frac{1}{3}\ln(8)\) 163. \(6 \log _{3} x+9 \log _{3} y\) 164. \(4 \log _{2} x-6 \log _{2} y\) 165. \(\frac{1}{3} \log _{2} x+\frac{2}{3} \log _{2} y\) 166. \(-2 \log _{5} x+\frac{3}{5} \log _{5} y\) 167. \(\log 5+3 \log (x+y)\) 168. \(4 \log _{5}(x+5)+\log _{5} y \)

169. \(2 \log (2 x+3)+\frac{1}{2} \log (x+1)\) 170. \(\frac{1}{3} \log x-3 \log (x+1)\) 171. \(\frac{2}{3} \log _{3} x-\frac{1}{2}\left(\log _{3} y+\log _{3} z\right)\) 172. \(\log _{3}\left(x^{2}-1\right)-2 \log _{3}(x-1)\) 173. \(\log \left(x^{2}+2 x+1\right)-2 \log (x+1)\) 178. \(\frac{1}{5}\left(\log _{7} x+2 \log _{7} y\right)-2 \log _{7}(z+1)\) 174) \(2\log (x)+3\log (x+1)\) 175. \(\frac{1}{3}(\ln x+2 \ln y)-(3 \ln 2+\ln z)\)

Answers to odd exercises:

161. \(\log_b(7)\) 163. \(\log _{3} x^{6} y^{9}\)

165. \(\log _{2}\left(\sqrt[3]{x y^{2}}\right)\) 167. \(\log \left(5(x+y)^{3}\right)\)

169. \(\log (2 x+3)^{2} \cdot \sqrt{x+1}\) 171. \(\log _{3}\left(\dfrac{\sqrt[3]{x^{2}}}{\sqrt{y z}}\right)\)

173. \(0\) 175. \(\ln \left(\dfrac{\sqrt[3]{x y^{2}}}{8 z}\right)\)

 \( \bigstar \) For the following exercises, condense each expression to a single logarithm with a coefficient \(1\) using the properties of logarithms.

176. \(4\log _7(c)+\frac{\log _7(a)}{3}+\frac{\log _7(b)}{3}\) 177. \(3 \ln x+4 \ln y-2 \ln z\) 178. \(4 \log x-2 \log y-3 \log z\) 179. \(\log _{2} 5+2 \log _{2} x+\log _{2} y\) 180. \(\log _{3} 4+3 \log _{3} x+\frac{1}{2} \log _{3} y\) 181. \(3 \log _{2} x-2 \log _{2} y+\frac{1}{2} \log _{2} z\) 182. \(4 \log x-\log y-\log 2\) 183. \(\ln x-6 \ln y+\ln z\) 184. \(\log _{3} x-2 \log _{3} y+5 \log _{3} z\) 185. \(7 \log x-\log y-2 \log z\)

186. \(2 \ln x-3 \ln y-\ln z\) 187. \(1+\log _{2} x-\frac{1}{2} \log _{2} y\) 188. \(2-3 \log _{3} x+\frac{1}{3} \log _{3} y\) 189. \(-\ln 2+2 \ln (x+y)-\ln z\) 190. \(-3 \ln (x-y)-\ln z+\ln 5\) 191. \(\log (x)-\frac{1}{2}\log (y)+3\log (z)\) 192. \(4 \log 2+\frac{2}{3} \log x-4 \log (y+z)\) 193. \(\log _{2} 3-2 \log _{2} x+\frac{1}{2} \log _{2} y-4 \log _{2} z\) 194. \(2 \log _{5} 4-\log _{5} x-3 \log _{5} y+\frac{2}{3} \log _{5} z\)

Answers to odd exercises:

177. \(\ln \frac{x^{3} y^{4}}{z^{2}}\) 179. \(\log _{2}\left(5 x^{2} y\right)\)

181. \(\log _{2}\left(\frac{x^{3} \sqrt{z}}{y^{2}}\right)\) 183. \(\ln \left(\frac{x z}{y^{6}}\right)\)

185. \(\log \left(\frac{x^{7}}{y z^{2}}\right)\) 187. \(\log _{2}\left(\frac{2 x}{\sqrt{y}}\right)\)

189. \(\ln \left(\dfrac{(x+y)^{2}}{2 z}\right)\) 191. \(\log \left ( \dfrac{xz^3}{\sqrt{y}} \right )\)

193. \(\log _{2}\left(\frac{3 \sqrt{y}}{x^{2} z^{4}}\right)\)

G: Use the Change of Base Formula

Exercise \(\PageIndex{G}\) 

\( \bigstar \) For the following exercises, rewrite each expression as an equivalent ratio of logs using the indicated base.

197. \(\log _7(15)\) to base \(e\)

198. \(\log _{14}(55.875)\) to base \(10\)

\( \bigstar \) For the following exercises, use the change-of-base formula and either base 10 or base e to evaluate the given expressions. Answer in exact form and in approximate form, rounding to four decimal places.

199. \(\log _3 (22)\) 200. \(\log _8 (65)\) 201. \(\log _6 (5.38)\)

202. \(\log _4 \left (\frac{15}{2} \right )\) 203. \(\log _{\frac{1}{2}} (4.7)\) 204. \(\log _{3} 42\)

205. \(\log _{5} 46\) 206. \(\log _{12} 87\) 207. \(\log _{15} 93\)

208. \(\log _{\sqrt{2}} 17\) 209. \(\log _{\sqrt{3}} 21\) 210. \(\log_547\)

211. \(\log_782\) 212. \(\log_6103\) 213. \(\log_{0.5}211\)

214. \(\log_2π\) 215. \(\log_{0.2}0.452\)

\( \bigstar \) For the following exercises, suppose \(\log _5(6)=a\) and \(\log _5(11)=b\). Use the change-of-base formula along with properties of logarithms to rewrite each expression in terms of \(a\) and \(b\). Show the steps for solving.

216. \(\log _{11} (5)\)

217. \(\log _{6} (55)\)

218. \(\log _{11}\left (\dfrac{6}{11} \right )\)

Answers to odd exercises:

197. \(\dfrac{\ln (15)}{\ln (7)} \approx 1.3917 \) 199. \( \dfrac{ \ln(22)}{ \ln(3)} \approx 2.8136\) 201.\(\dfrac{ \ln(5.38)}{ \ln(6)} \approx 0.9391\)

203. \( \dfrac{ \ln(4.7)}{ \ln(.5)} \approx -2.2327\) 205. \(\dfrac{ \ln(46)}{ \ln(5)} \approx 2.3789\) 207. \( \dfrac{ \ln(93)}{ \ln(15)} \approx 1.6738\)

209. \( \dfrac{ \ln(21)}{ \ln( \sqrt{3}} \approx 5.5425\) 211. \(\dfrac{\log82}{\log7}≈2.2646\) 213. \(\dfrac{\log211}{\log0.5}≈−7.7211\)

215. \(\dfrac{\log0.452}{\log0.2}≈0.4934\) 217. \(\dfrac{\log_5 (5\cdot 11)}{\log_5 (6)}=\dfrac{1+b}{a}\)

H: "Extensions"

Exercise \(\PageIndex{H}\) 

220. Use the product rule for logarithms to find all \(x\) values such that \(\log _{12} (2x+6)+\log _{12} (x+2)=2\)$\mathrm{.}$

221. Use the quotient rule for logarithms to find all \(x\) values such that \(\log _{6} (x+2)-\log _{6} (x-3)=1\)$\mathrm{.}$

222. Can the power property of logarithms be derived from the power property of exponents using the equation \(b^x=m\)$?$If not, explain why. If so, show the derivation.

223. Prove that \(\log_b(n)=\frac{1}{\log_n(b)}\) for any positive integers \(b>1\) and \(n>1\)$\mathrm{.}$

224. Does \(\log_{81}(2401)=\log_3(7)\) ?   Verify the claim algebraically.

Answers  to odd exercises:

221. Rewriting as an exponential equation and solving for \(x\): \( 6^1 = \frac{x+2}{x-3} \rightarrow 0 = \frac{x+2}{x-3}-6= \frac{x+2}{x-3}-\frac{6(x-3)}{(x-3)} = \frac{x+2-6x+18}{x-3} = \frac{x-4}{x-3} \rightarrow x = 4\).  Checking, we find that \(\log _6(4+2)-\log _6(4-3)=\log _6(6)-\log _6(1)\) is defined, so \(x=4\)

223. Let \(b\) and \(n\) be positive integers \( > 1\) .  By the change-of-base formula, \(\log_b(n)=\frac{\log_n(n)}{\log_n(b)}=\frac{1}{\log_n(b)}\)