Using the Quotient RuleUse the quotient rule to rewrite each logarithm. Assume x 7 0.
EXAMPLE 2
SECTION 10.4 Properties of Logarithms 597
CAUTION There is no property of logarithms to rewrite the logarithm of a
sum or difference.For example, we cannotwrite in terms of and
Also,
logb
x
y
Z
logb x
logb y
.
logb y.
logb 1 x+y 2 logb x
(a)
= log 4 7 - log 4 9 Quotient rule
log 4
7
9
(b)
= log 5 Quotient rule
6
x
log 5 6 - log 5 x
(c)
Quotient rule= 3 - log 3 5 log 3 27 = 3 NOW TRY
= log 3 27 - log 3 5
log 3
27
5
OBJECTIVE 3 Use the power rule for logarithms. An exponential expression
such as means The base is used as a factor 3 times. Similarly, the prod-
uct rule can be extended to rewrite the logarithm of a power as the product of the ex-
ponent and the logarithm of the base.
Furthermore, we saw in Example 1(d)that These examples sug-
gest the following rule.
log 4 x^3 =3 log 4 x.
= 3 log 5 2 =4 log 2 7
= log 5 2 + log 5 2 + log 5 2 =log 2 7 +log 2 7 +log 2 7 + log 2 7
= log 5 12 # 2 # 22 =log 2 17 # 7 # 7 # 72
log 5 23 log 2 74
23 2 # 2 #2.
Power Rule for LogarithmsIf xand bare positive real numbers, where and if ris any real number,
then the following is true.
That is, the logarithm of a number to a power equals the exponent times the loga-
rithm of the number.
logb xrr logb x
bZ 1,
As further examples of this rule,
logb m^5 = 5 logb m and log 3 54 = 4 log 3 5.
NOW TRY
EXERCISE 2
Use the quotient rule to
rewrite each logarithm.
(a)
(b)
(c) log 5
25
27
log 4 x-log 4 12, x 70
log 107
9
NOW TRY ANSWERS
- (a)
(b)
(c) 2 - log 5 27log 4 12 xlog 10 7 - log 10 9