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4.3E: Exercises - Logarithm Functions

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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.

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) $$\ln(w)=n$$
 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) $$\sqrt{64}=4$$ 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. $$x^{\frac{1}{3}}=\sqrt{6}$$ 64. $$32^{x}=\sqrt{32}$$ 65. $$17^{x}=\sqrt{17}$$ 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) $$c^d=k$$
 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. $$\log _{x} \sqrt{6}=\frac{1}{3}$$ 65. $$\log _{17} \sqrt{17}=x$$ 67. $$\ln x=3$$ 69. $$\ln(w)=n$$ 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. $$\log _{5} \sqrt{5}$$ 104. $$\log _{2} \sqrt{2}$$ 105. $$\log _{7}\left(\frac{1}{\sqrt{7}}\right)$$ 106. $$\log _{9}\left(\frac{1}{\sqrt{9}}\right)$$ 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. $$\log _{3 / 5}\left(\frac{5}{3}\right)$$
 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) $$\ln \left ( e^{-0.225} \right )-3$$
 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) $$\ln (\sqrt{2})$$
 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. $$\log _{2} \sqrt{2}=x$$ 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. $$\log _{x} 27=3$$
 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. $$\sqrt{e}$$ 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. $$\ln x=-1$$
 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:

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.

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. $$2$$
$$\star$$