Intermediate Algebra (11th edition)

NOTE The word statement of the product rule can be restated by replacing “loga-

rithm” with “exponent.” The rule then becomes the familiar rule for multiplying ex-

ponential expressions: The exponentof a product is the sum of the exponentsof the

factors.

596 CHAPTER 10 Inverse, Exponential, and Logarithmic Functions

To prove this rule, let and and recall that

means and means

Now consider the product xy.

Substitute.

Product rule for exponents

Convert to logarithmic form.

Substitute.

The last statement is the result we wished to prove.

logb xy=logb x+ logb y

logb xy=m+ n

xy=bm+n

xy=bm #bn

logb x= m bm= x logb y= n bn= y.

m= logb x n=logb y,

Using the Product Rule

Use the product rule to rewrite each logarithm. Assume x 7 0.

EXAMPLE 1

(a)

= log 5 6 + log 5 9 Product rule

log 5 16 # 92

(c)

Product rule

(d)

Product rule

=3 log 4 x Combine like terms. NOW TRY

= log 4 x+log 4 x+log 4 x

= log 4 1 x#x#x 2 x^3 =x#x#x

log 4 x^3

= 1 + log 3 x log 3 3 = 1

= log 3 3 + log 3 x

log 3 13 x 2

OBJECTIVE 2 Use the quotient rule for logarithms. The rule for division is

similar to the rule for multiplication.

Quotient Rule for Logarithms

If x, y, and bare positive real numbers, where then the following is true.

That is, the logarithm of a quotient is the difference between the logarithm of the

numerator and the logarithm of the denominator.

logb

x

y

logb xlogb y

bZ1,

The proof of this rule is similar to the proof of the product rule.

(b)

Product rule

= log 7 96 Multiply.

= log 7 18 # 122

log 7 8 +log 7 12

NOW TRY
EXERCISE 1
Use the product rule to
rewrite each logarithm.

(a)

(b)

(c)

(d)log 2 t^3 , t 70

log 5 15 x 2 , x 70

log 5 11 +log 5 8

log 10 17 # 92

NOW TRY ANSWERS

1. (a)
   (b)
   (c)
   (d)3 log 2 t

1 +log 5 x

log 5 88

log 10 7 +log 10 9