Solutions: Logarithmic Functions

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 byby can also be written as the logarithm, logbxlogbx, 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=xby=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)loge(x), the notation used is ln(x)ln(x).

7. ac=bac=b

9. xy=64xy=64

11. 15b=a15b=a

13. 13a=14213a=142

15. en=wen=w

17. logc(k)=dlogc(k)=d

19. log19y=xlog19y=x

21. logn(103)=4logn(103)=4

23. logy(39100)=xlogy(39100)=x

25. ln(h)=kln(h)=k

27. x=23=18x=23=18

29. x=33=27x=33=27

31. x=912=3x=912=3

33. x=63=1216x=63=1216

35. x=e2x=e2

37. 32

39. 1.06

41. 14.125

43. 1212

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 = 0. To verify, suppose = 0 is in the domain of the function f(x)=log(x)f(x)=log(x). Then there is some number n such that n=log(0)n=log(0). Rewriting as an exponential equation gives: 10n=010n=0, which is impossible since no such real number n exists. Therefore, = 0 is not the domain of the function f(x)=log(x)f(x)=log(x).

61. Yes. Suppose there exists a real number x such that lnx=2lnx=2. Rewriting as an exponential equation gives x=e2x=e2, which is a real number. To verify, let x=e2x=e2. Then, by definition, ln(x)=ln(e2)=2ln(x)=ln(e2)=2.

63. No; ln(1)=0ln(1)=0, so ln(e1.725)ln(1)ln(e1.725)ln(1) is undefined.

65. 2