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Mathematics LibreTexts

4.5e: Exercises - Properties of Logarithms

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    44987
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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?
    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. \(\log _{5} 8+\log _{5} y\)

    52. \(4+\log _{3} x+\log _{3} y\)

    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. \(\log _b 2+\log _b 7+\log _b (x)+\log _b (y)\)

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

    127. \(2 \log _{3} x+\frac{1}{3} \log _{3} y-\log _{3} z\)

    129. \(3 \log _{5} x-\frac{1}{3} \log _{5} y-\frac{2}{3} \log _{5} z \\[6pt] \)

    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. \(\log _{2} 5-\log _{2}(x-1)\)

    15-. \(\log _{3} 4+\log _{3}(x+1)\)

    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 \(\PageIndex{H}\) 

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

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

    220) 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.

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

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

    Answers  to odd exercises:

    219. 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 \(x=4\)

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

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