1. a. log10(1,000,000)=6 is equivalent to 106=1,000,000
b. log5(25)=2 is equivalent to 52=25
2. a. 32=9 is equivalent to log3(9)=2
b. 53=125 is equivalent to log5(125)=3
c. 2−1=21 is equivalent to log2(21)=−1
3. log121(11)=21 (recalling that 121=(121)21=11 )
4. log2(321)=−5
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 by can also be written as the logarithm, logbx, 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 by=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 loge(x), the notation used is ln(x).
7. ac=b
9. xy=64
11. 15b=a
13. 13a=142
15. en=w
17. logc(k)=d
19. log19y=x
21. logn(103)=4
23. logy(10039)=x
25. ln(h)=k
27. x=2−3=81
29. x=33=27
31. x=921=3
33. x=6−3=2161
35. x=e2
37. 32
39. 1.06
41. 14.125
43. 21
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 f(x)=log(x). Then there is some number n such that n=log(0). Rewriting as an exponential equation gives: 10n=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).
61. Yes. Suppose there exists a real number x such that lnx=2. Rewriting as an exponential equation gives x=e2, which is a real number. To verify, let x=e2. Then, by definition, ln(x)=ln(e2)=2.
63. No; ln(1)=0, so ln(1)ln(e1.725) is undefined.
65. 2
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Precalculus.Provided by: OpenStaxAuthored by: Jay Abramson, et al..Located at: https://openstax.org/books/precalculus/pages/1-introduction-to-functions.License: CC BY: Attribution. License terms: Download For Free at : http://cnx.org/contents/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175..