SOLUTIONS TO TOOLS FOR CHAPTER 4
261
21. Rewrite the logarithm of the product as a sum, log 2x = log 2 + log x.
22. Rewrite the expression as a sum and then use the power property,
ln(x(7 − x)3 ) = ln x + ln(7 − x)3 = ln x + 3 ln(7 − x).
23. The expression is not the logarithm of a quotient, so it cannot be rewritten using the properties of logarithms.
24. The logarithm of a quotient rule applies, so log
x
5
= log x − log 5.
25. The logarithm of a quotient and the power property apply, so
log
x2 + 1
x3
26. Rewrite the power,
ln
Use the quotient and power properties,
ln
r
= log(x2 + 1) − log x3
= log(x2 + 1) − 3 log x.
(x − 1)1/2
x−1
.
= ln
x+1
(x + 1)1/2
(x − 1)1/2
= ln(x − 1)1/2 − ln(x + 1)1/2 = (1/2) ln(x − 1) − (1/2) ln(x + 1).
(x + 1)1/2
27. There is no rule for the logarithm of a sum, it cannot be rewritten.
28. Using the properties of logarithms we have
ln
xy 2
z
= ln xy 2 − ln z
= ln x + ln y 2 − ln z
= ln x + 2 ln y − ln z.
29. In general the logarithm of a difference cannot be simplified. In this case we rewrite the expression so that it is the
logarithm of a product.
log(x2 − y 2 ) = log((x + y)(x − y)) = log(x + y) + log(x − y).
30. The expression is a product of logarithms, not a logarithm of a product, so it cannot be simplified.
31. The expression is a quotient of logarithms, not a logarithm of a quotient, but we can use the properties of logarithms to
rewrite it without the x2 term:
ln x2
2 ln x
=
.
ln(x + 2)
ln(x + 2)
32. Rewrite the sum as log 12 + log x = log 12x.
33. Rewrite the sum as ln x3 + ln x2 = ln(x3 · x2 ) = ln x5 .
34. Rewrite the difference as
ln x2 − ln(x + 10) = ln
x2
.
x + 10
262
Chapter Four /SOLUTIONS
35. Rewrite with powers and combine,
√
√
1
log x + 4 log y = log x + log y 4 = log( xy 4 ).
2
36. Rewrite with powers and combine,
log 3 + 2 log
√
√
x = log 3 + log( x)2 = log 3 + log x = log 3x.
37. Rewrite with powers and combine,
1
1
log 8 − log 25 = log 81/3 − log 251/2
3
2
= log 2 − log 5
2
= log .
5
38. Rewrite with powers and combine,
3 log(x + 1) +
2
log(x + 4) = 3 log(x + 1) + 2 log(x + 4)
3
= log(x + 1)3 + log(x + 4)2
= log (x + 1)3 (x + 4)2
39. Rewrite as
ln x + ln
y
xy + 4y
(x + 4) + ln z −1 = ln x + ln
2
2
− ln z = ln
(x2 )y + 4xy
2
− ln z = ln
40. Rewrite with powers and combine,
2 log(9 − x2 ) − (log(3 + x) + log(3 − x)) = log(9 − x2 )2 − (log(3 + x)(3 − x))
= log(9 − x2 )2 − log(9 − x2 )
= log
(9 − x2 )2
(9 − x2 )
= log(9 − x2 ).
41. Rewrite as 10− log 5x = 10log(5x)
−1
= (5x)−1 .
42. Rewrite as e−3 ln t = eln t = t−3 .
√
√
43. Rewrite as 2 ln e x = 2 x.
−3
44. The logarithm of a sum cannot be simplified.
45. Rewrite as t ln et/2 = t(t/2) = t2 /2.
46. Rewrite as 102+log x = 102 · 10log x = 100x.
47. Rewrite as log(10x) − log x = log(10x/x) = log 10 = 1.
48. Rewrite as 2 ln x−2 + ln x4 = 2(−2) ln x + 4 ln x = 0.
√
49. Rewrite as ln x2 + 16 = ln(x2 + 16)1/2 = 21 ln(x2 + 16).
50. Rewrite as log 1002z = 2z log 100 = 2z(2) = 4z.
(x2 )y + 4xy
2z
.