Solutions to Try Its
1. a. [latex]{\mathrm{log}}_{10}\left(1,000,000\right)=6[/latex] is equivalent to [latex]{10}^{6}=1,000,000[/latex]
b. [latex]{\mathrm{log}}_{5}\left(25\right)=2[/latex] is equivalent to [latex]{5}^{2}=25[/latex]
2. a. [latex]{3}^{2}=9[/latex] is equivalent to [latex]{\mathrm{log}}_{3}\left(9\right)=2[/latex]
b. [latex]{5}^{3}=125[/latex] is equivalent to [latex]{\mathrm{log}}_{5}\left(125\right)=3[/latex]
c. [latex]{2}^{-1}=\frac{1}{2}[/latex] is equivalent to [latex]{\text{log}}_{2}\left(\frac{1}{2}\right)=-1[/latex]
3. [latex]{\mathrm{log}}_{121}\left(11\right)=\frac{1}{2}[/latex] (recalling that [latex]\sqrt{121}={\left(121\right)}^{\frac{1}{2}}=11[/latex] )
4. [latex]{\mathrm{log}}_{2}\left(\frac{1}{32}\right)=-5[/latex]
5. It is not possible to take the logarithm of a negative number in the set of real numbers.
6. It is not possible to take the logarithm of a negative number in the set of real numbers.
Solutions to Odd-Numbered 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 [latex]{b}^{y}[/latex] can also be written as the logarithm, [latex]{\mathrm{log}}_{b}x[/latex], 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 [latex]{b}^{y}=x[/latex], 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 [latex]{\mathrm{log}}_{e}\left(x\right)[/latex], the notation used is [latex]\mathrm{ln}\left(x\right)[/latex].
7. [latex]{a}^{c}=b[/latex]
9. [latex]{x}^{y}=64[/latex]
11. [latex]{15}^{b}=a[/latex]
13. [latex]{13}^{a}=142[/latex]
15. [latex]{e}^{n}=w[/latex]
17. [latex]{\text{log}}_{c}\left(k\right)=d[/latex]
19. [latex]{\mathrm{log}}_{19}y=x[/latex]
21. [latex]{\mathrm{log}}_{n}\left(103\right)=4[/latex]
23. [latex]{\mathrm{log}}_{y}\left(\frac{39}{100}\right)=x[/latex]
25. [latex]\text{ln}\left(h\right)=k[/latex]
27. [latex]x={2}^{-3}=\frac{1}{8}[/latex]
29. [latex]x={3}^{3}=27[/latex]
31. [latex]x={9}^{\frac{1}{2}}=3[/latex]
33. [latex]x={6}^{-3}=\frac{1}{216}[/latex]
35. [latex]x={e}^{2}[/latex]
37. 32
39. 1.06
41. 14.125
43. [latex]\frac{1}{2}[/latex]
45. 4
47. –3
49. –12
51. 0
53. 10
55. 2.708
57. 0.151
59. No, the function has no defined value for x = 0. To verify, suppose x = 0 is in the domain of the function [latex]f\left(x\right)=\mathrm{log}\left(x\right)[/latex]. Then there is some number n such that [latex]n=\mathrm{log}\left(0\right)[/latex]. Rewriting as an exponential equation gives: [latex]{10}^{n}=0[/latex], which is impossible since no such real number n exists. Therefore, x = 0 is not the domain of the function [latex]f\left(x\right)=\mathrm{log}\left(x\right)[/latex].
61. Yes. Suppose there exists a real number x such that [latex]\mathrm{ln}x=2[/latex]. Rewriting as an exponential equation gives [latex]x={e}^{2}[/latex], which is a real number. To verify, let [latex]x={e}^{2}[/latex]. Then, by definition, [latex]\mathrm{ln}\left(x\right)=\mathrm{ln}\left({e}^{2}\right)=2[/latex].
63. No; [latex]\mathrm{ln}\left(1\right)=0[/latex], so [latex]\frac{\mathrm{ln}\left({e}^{1.725}\right)}{\mathrm{ln}\left(1\right)}[/latex] is undefined.
65. 2