4.3e: Exercises - Logarithm Functions

A: Concepts

Exercise \(\PageIndex{A}\) 

1) What is a base \(b\) logarithm? Discuss the meaning by interpreting each part of the equivalent equations \(b^y=x\) and \(\log _bx=y\) for \(b>0, b\neq 1\)

2) How is the logarithmic function \(f(x)=\log _bx\) related to the exponential function \(g(x)=b^x\)? What is the result of composing these two functions?

3) How can the logarithmic equation \(\log _bx=y\) be solved for \(x\) using the properties of exponents?

4) Discuss the meaning of the common logarithm. What is its relationship to a logarithm with base \(b\), and how does the notation differ?

5) Discuss the meaning of the natural logarithm. What is its relationship to a logarithm with base \(b\), and how does the notation differ?

6) Is \(f(x)=0\) in the range of the function \(f(x)=\log (x)\)?   If so, for what value of \(x\)?   Verify the result.

7) Is \(x=0\)  in the domain of the function \(f(x)=\log x\)? If so, what is the value of the function when \(x=0\)? Verify the result.

Answers to odd exercises:

1. A logarithm is an exponent. Specifically, it is the exponent to which a base \(b\) is raised to produce a given value. In the expressions given, the base \(b\) has the same value. The exponent, \(y\), in the expression \(b^y\) can also be written as the logarithm, \(\log _bx=y\), and the value of \(x\) is the result of raising \(b\) to the power of \(y\).

3. Since the equation of a logarithm is equivalent to an exponential equation, the logarithm can be converted to the exponential equation \(b^y = x\) , and then properties of exponents can be applied to solve for \(x\) .

5. The natural logarithm is a special case of the logarithm with base \(b\) in that the natural log always has base \(e\).   Rather than notating the natural logarithm as \(\log_{e}(x)\) , the notation used is \(\ln (x)\).

7. No, the function has no defined value for \(x=0\) .   To verify, suppose \(x=0\) is in the domain of the function \(f(x)=\log (x)\) .   Then there is some number \(n\) such that \(n=\log(0)\) .   Rewriting as an exponential equation gives: \(10^n=0\) , which is impossible since no such real number \(n\) exists. Therefore, \(x=0\) is not the domain of the function \(f(x)=\log (x)\).

B: Convert from log to exponential form

Exercise \(\PageIndex{B}\) 

\( \bigstar \) For the following exercises, rewrite each equation in exponential form.

8) \(\log_381=4\) 9) \(\log_82=\frac{1}{3}\) 10) \(\log_51=0\) 11) \(\log_525=2\) 12) \(\log 0.1=−1\) 13) \(\log_9 3=0.5\)

14. \(3=\log _{4} 64\) 15. \(6=\log _{2} 64\) 16. \(0=\log _{12} 1\) 17. \(0=\log _{7} 1\) 18. \(1=\log _{3} 3\) 19. \(1=\log _{9} 9\)

20. \(5=\ln x\) 21) \(\ln1=0\) 22) \(\ln (\frac{1}{e^3})=−3\) 23. \(x=\ln 43\) 24. \(-4=\log \frac{1}{10,000}\) 25. \(3=\log  1,000\)

26. \(4=\log _{x} 81\) 27. \(5=\log _{x} 32\) 28) \(\log_{y}(x)=-11\) 29) \(\log_{13}(142)=a\) 30) \(\log_{y}(137)=x\) 31) \(\log_{x}(64)=y\)

32) \(\log_{4}(q)=m\) 33) \(\log_{15}(a)=b\) 34) \(\log_{16}(y)=x\) 35) \(\log_{a}(b)=c\) 36) \(\log(v)=t\) 37) \(\ln(w)=n\)

Answers to odd exercises:

9: \(8^{1/3}=2\) 11: \(5^2=25\) 13. \( 9^{0.5} = 3 \)

15. \(64=2^{6}\) 17. \(1=7^{0}\) 19. \(9=9^{1}\)

21: \(e^0=1\) 23. \(43=e^{x}\) 25. \(1,000=10^{3}\)

27. \(32=x^{5}\) 29. \(13^a=142\) 31. \(x^y=64\)

33. \(15^b=a\) 35. \(a^c=b\) 37. \(e^n=w\)

C: Convert from exponential to log form

Exercise \(\PageIndex{C}\) 

\( \bigstar \) For the following exercises, write the equation in equivalent logarithmic form. 

38. \(2^3=8\) 39. \(4^{−2}=\frac{1}{16}\) 40. \(10^2=100\) 41. \(9^0=1\) 42. \((\frac{1}{3})^3=\frac{1}{27}\) 43. \(4^{-3/2}=0.125\) 44. \(\sqrt[3]{64}=4\) 45. \(n^4 = 103\)

46. \(4^{2}=16\) 47. \(2^{5}=32\) 48. \(3^{3}=27\) 49. \(5^{3}=125\) 50. \(b^3=45\) 51. \(9^y=150\) 52. \(10^{3}=1000\) 53. \(10^{-2}=\frac{1}{100}\)

58. \(\left(\frac{1}{4}\right)^{2}=\frac{1}{16}\) 59. \(\left(\frac{1}{3}\right)^{4}=\frac{1}{81}\) 60. \(3^{-2}=\frac{1}{9}\) 61. \(4^{-3}=\frac{1}{64}\) 62. \(x^{\frac{1}{2}}=\sqrt{3}\) 63. \(x^{\frac{1}{3}}=\sqrt[3]{6}\)

64. \(32^{x}=\sqrt[4]{32}\) 65. \(17^{x}=\sqrt[5]{17}\) 66. \(e^{x}=6\) 67. \(e^{3}=x\) 68. \(e^x=y\) 69. \(e^k=h\)

70. \(m^{-7}=n\) 71. \(10^a=b\) 72. \(4^x=y\) 73. \(19^x=y\) 74. \(x^{-\frac{10}{13}}=y\) 75. \(y^x=\frac{39}{100}\) 76. \(\left ( \frac{7}{5} \right )^m=n\) 77. \(c^d=k\)

Answers to odd exercises:

39: \(\log_4(\frac{1}{16})=−2\) 41: \(\log_91=0\) 43: \(\log_40.125=−\frac{3}{2}\) 45. \(\log_{n}(103)=4\)

47. \(\log _{2} 32=5\) 49. \(\log _{5} 125=3\) 51: \(\log_9150=y\) 53. \(\log \frac{1}{100}=-2\)

59. \(\log _{\frac{1}{3}} \frac{1}{81}=4\) 61. \(\log _{4} \frac{1}{64}=-3\) 63. \(\log _{x} \sqrt[3]{6}=\frac{1}{3}\)

65. \(\log _{17} \sqrt[5]{17}=x\) 67. \(\ln x=3\) 69. \( k = \ln(h) \)

71. \(\log (b)=a\) 73. \(\log_{19}(y)=x\) 75. \(\log_{y}\left ( \frac{39}{100} \right )=x\) 77. \(\log_{c}(k)=d\)

D: Evaluate logarithms using the definition

Exercise \(\PageIndex{D}\) 

\( \bigstar \) In the following exercises, find the exact value of each logarithm without using a calculator.

78. \(\log _{3} 243\) 79. \(\log _{3} 9\) 80. \(\log _{4} 4\) 81. \(\log _{5} 1\) 82. \(\log _{5} 625\) 83. \(\log _{6} 36\) 84. \(\log _{7} 49\)

85. \(\log _{25} 5\) 86. \(\log _{8} 2\) 87. \(\log _{27} 3\) 88. \(\log _{16} 4\) 89. \(\log _{4} 4^{10}\) 90. \(\log _{9} 9^{5}\) 91) \(6\log _8(4)\)

92. \(\log _{2}\left(\frac{1}{64}\right)\) 93. \(\log _{2}\left(\frac{1}{16}\right)\) 94) \(\log _2\left ( \frac{1}{8} \right )+4\) 95. \(\log _{3} \frac{1}{27}\) 96. \(\log _{3}\left(\frac{1}{9}\right)\) 97. \(\log _{4}\left(\frac{1}{2}\right)\) 98. \(\log _{4} \frac{1}{16}\)

100. \(\log _{5}\left(\frac{1}{125}\right)\) 101. \(\log _{9} \frac{1}{81}\) 102) \(\log _6(\sqrt{6})\) 103. \(\log _{5} \sqrt[3]{5}\) 104. \(\log _{2} \sqrt{2}\) 105. \(\log _{7}\left(\frac{1}{\sqrt{7}}\right)\) 106. \(\log _{9}\left(\frac{1}{\sqrt[3]{9}}\right)\) 107. \(\log _{1 / 9} 1\)

108. \(\log _{1 / 3} 27\) 109. \(\log _{1 / 2} 4\) 110. \(\log _{\frac{1}{2}} 2\) 111. \(\log _{\frac{1}{2}} 4\) 112. \(\log _{3 / 4}\left(\frac{9}{16}\right)\) 113. \(\log _{2 / 3}\left(\frac{2}{3}\right)\) 114. \(\log _{27}\left(\frac{1}{3}\right)\) 115. \(\log _{3 / 5}\left(\frac{5}{3}\right)\)

Answers to odd exercises:

79. \(2\) 81. \(0\) 83. \(2\)

85. \(\frac{1}{2}\) 87. \(\frac{1}{3}\) 89. \(10\) 91. \(4\)

93. \(−4\) 95. \(-3\) 97. \(−\frac{1}{2}\)

101. \(-2\) 103. \(\frac{1}{3}\) 106. \(−\frac{1}{2}\) 107. \(0\)

109. \(−2\) 111. \(-2\) 113. \(1\) 115. \(-1\)

\( \bigstar \)

E: Evaluate common and natural logarithms without a calculator

Exercise \(\PageIndex{E}\) 

\( \bigstar \) For the following exercises, use the definition of common and natural logarithms to evaluate without using a calculator.

117. \(\log 1000\) 118. \(\log 100\) 119) \(\log (10,000)\) 120) \(\log (100^8)\)

121.  \(\log 0.1\)   122) \(2\log (.0001)\) 123) \(\log (0.001)\) 124) \(\log (1)+7\)

125) \(2\log (100^{-3})\) 126) \(10^{\log (32)}\) 127) \(e^{\ln (1.06)}\) 128) \(e^{\ln (10.125)}+4\)

129. \(\ln \left(\frac{1}{e}\right)\) 130. \(\ln \left(\frac{1}{e^{5}}\right)\) 131) \(25\ln \left ( e^{\frac{2}{5}} \right )\) 132) \(\ln \left ( e^{\frac{1}{3}} \right )\)

133. \(\ln e^{4}\) 134) \(\ln (e^{-5.03})\) 135) \(\ln (1)\) 136) \(\ln \left ( e^{-0.225} \right )-3\)

Answers to odd exercises:

117. \(3\) 119. \(4\)

12`. \(−1\) 123. \(-3\)

125. \(-12\) 127. \(1.06\)

129. \(−1\) 131. \(10\)

133. \(4\) 135. \(0\)

F: Use a calculator to evaluate logs

Exercise \(\PageIndex{F}\) 

\( \bigstar \) For the following exercises, evaluate each expression using a calculator. Round to the nearest thousandth.

139. \(\log 162\) 140. \(\log e\)

141. \(\log 0.025\) 142. \(\log 0.235\)

143. \(\ln (25)\) 144. \(\ln (100)\)

145. \(\ln (0.125)\) 146. \(\ln (0.001)\)

147) \(\ln (15)\) 148) \(\ln \left ( {\frac{4}{5}} \right )\)

149) \(\log (\sqrt{2})\) 150) \(\ln (\sqrt{2})\)

Answers to odd exercises:

139. \(2.210\)

141. \(−1.602\)

143. \(3.219\)

145. \(−2.079\)

147. \(2.708\)

149. \(0.151\)

G: Solve log equations by converting to exponential form first

Exercise \(\PageIndex{G}\) 

\( \bigstar \) In the following exercises, find the value of \(x\) in each logarithmic equation without using a calculator by first converting the logarithmic equation to exponential form.

151) \(\log_{2}(x)=-3\) 152. \(\log _{2} (x)=-6\) 153) \(\log_{2}(x)=6\) 154. \(\log _{2} (x)=5\) 155. \(\log _{2} (x)=8\) 156. \(\log _{2} \sqrt[5]{2}=x\) 157) \(\log_{3}(x)=3\) 158. \(\log _{3} (x)=-5\)

159. \(\log _{3} (x)=4\) 160.  \(\log _{3}\left(\frac{1}{27}\right)=x\) 161. \(\log _{5} (x)=-3\) 162) \(\log_{5}(x)=2\) 163. \(\log _{5} (x)=3\) 164. \(\log _{7} (x)=-1\) 165)  \(\log_{6}(x)=-3\) 166. \(\log _{6} (x)=-2\)

167) \(\log_{9}(x)=\frac{1}{2}\) 168) \(\log_{18}(x)=2\) 169. \(\log _{12} (x)=0\) 170) \(\log (x)=3\) 171) \(\ln(x)=2\) 172. \(\ln (x)=9\) 173. \(\ln (x)=\frac{1}{5}\)

175. \(\log _{1 / 4} (x)=-2\) 176. \(\log _{2 / 5} (x)=2\) 177. \(\log _{1 / 9} (x)=\frac{1}{2}\) 178. \(\log _{1 / 4} (x)=\frac{3}{2}\) 179. \(\log _{1 / 3} (x)=-1\) 180. \(\log _{1 / 5} (x)=0\) 181. \(\log 10^{12}=x\) 182. \(\ln e=x\)

183. \(\log _{1 / 8}\left(\frac{1}{64}\right)=x\) 184. \(\log _{4/9}\left(\frac{2}{3}\right)=x\) 185. \(\log _{\frac{1}{3}} \frac{1}{9}=x\) 186. \(\log _{\frac{1}{4}} \frac{1}{16}=x\) 187. \(\log _{\frac{1}{9}} 81=x\) 188. \(\log _{\frac{1}{4}} 64=x\) 189. \(\log _{x} 121=2\) 190. \(\log _{x} 49=2\) 191. \(\log _{x} 64=3\) 192. \(\log _{x} 27=3\)

Answers  to odd exercises:

151. \(x=2^{-3}=\dfrac{1}{8}\) 153. \( 64 \) 155. \(256\) 157. \(x = 3^3 = 27\)

159. \(81\) 161. \(\frac{1}{125}\) 163. \(x=125\) 165. \(x=6^{-3}=\dfrac{1}{216}\)

167. \(x=9^{\frac{1}{2}}=3\) 168. \(1\) 171. \(x=e^2\) 173. \(\sqrt[5]{e}\)

175. \(16\) 177. \(\frac{1}{3}\) 179. \(3\) 181. \(12\)

183. \(2\) 185. \(x=2\) 187. \(x=-2\) 189. \(x=11\) 191. \(x=4\)

\( \bigstar \)

H: Solve log equations by converting then using a calculator

Exercise \(\PageIndex{H}\) 

\( \bigstar \) Find \(x\). Round off to the nearest hundredth.

195. \(\log x=2.5\) 196. \(\log x=1.8\)

197. \(\log x=-1.22\) 198. \(\log x=-0.8\)

199. \(\ln x=3.1\) 200. \(\ln x=1.01\)

201. \(\ln x=-0.69\) 202. \(\ln x=-1\)

Answers  to odd exercises:

195. \(316.23\)

197. \(0.06\)

199. \(22.20\)

201. \(0.50\)

I: Extensions

Exercise \(\PageIndex{I}\) 

203) Is there a number \(x\) such that \(\ln x = 2\)? If so, what is that number? Verify the result.

204) Is the following true: \(\frac{\log _3(27)}{\log _4\left ( \frac{1}{64} \right )}=-1\)? Verify the result.

205) Is the following true: \(\frac{\ln (e^{1.725})}{\ln (1)}=1.725\) Verify the result.

206) The exposure index \(EI\) for a \(35\) millimeter camera is a measurement of the amount of light that hits the film. It is determined by the equation \(EI=\log _2\left ( \frac{f^2}{t} \right )\) where \(f\) is the “f-stop” setting on the camera, and \(t\) is the exposure time in seconds. Suppose the f-stop setting is \(8\) and the desired exposure time is \(2\)seconds. What will the resulting exposure index be?

207) Refer to the previous exercise. Suppose the light meter on a camera indicates an \(EI\) of   \(-2\) , and the desired exposure time is \(16\) seconds. What should the f-stop setting be?

208) The intensity levels \(I\) of two earthquakes measured on a seismograph can be compared by the formula \(\log \left ( \frac{I_1}{I_2} \right )=M_1-M_2\) where \(M\) is the magnitude given by the Richter Scale. In August 2009, an earthquake of magnitude \(6.1\) hit Honshu, Japan. In March 2011, that same region experienced yet another, more devastating earthquake, this time with a magnitude of \(9.0\). How many times greater was the intensity of the 2011 earthquake? Round to the nearest whole number.

Answers to odd exercises:

203. Yes. Suppose there exists a real number \(x\) such that \(\ln x = 2\). Rewriting as an exponential equation gives \(x=e^2\) which is a real number. To verify, let \(x=e^2\). Then, by definition, \(\ln (x)=\ln \left ( e^2 \right ) = 2\).

205. No; \(\ln (1) =0\), so \(\frac{\ln (e^{1.725})}{\ln (1)}=1.725\) is undefined.

207. \(2\)

\( \star \)