# 4.5e: Exercises - Properties of Logarithms

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### 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?

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, $$\log _b \left ( x^{\frac{1}{n}} \right ) = \dfrac{1}{n}\log_{b}(x)$$.

### 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. $$\log _{a} 10=1$$
 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. $$\ln x^{\sqrt{3}}$$
 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. $$\ln (a\sqrt[3]{b}) \\[6pt]$$
 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. $$\ln \left ( \dfrac{a^{-2}}{b^{-4}c^{5}} \right ) \\[6pt]$$
 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. $$\log_4\dfrac{\sqrt[3]{xy}}{64} \\[6pt]$$
 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. $$\ln \left(1.4 e^{22}\right)$$
 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. $$\log (2 x+1)+\log (x-3) \\ -\log \left(2 x^{2}-5 x-3\right)$$
 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. $$\frac{1}{3}(\ln x+2 \ln y)-(3 \ln 2+\ln z)$$
 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. $$2 \log _{5} 4-\log _{5} x-3 \log _{5} y+\frac{2}{3} \log _{5} z$$
 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. $$\log _{11}\left (\dfrac{6}{11} \right )$$
 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 $$\log _{12} (2x+6)+\log _{12} (x+2)=2$$$.$

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

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 $$n>1$$$.$

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

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, $$\log_b(n)=\frac{\log_n(n)}{\log_n(b)}=\frac{1}{\log_n(b)}$$