Slide 10- 1Copyright © 2012 Pearson Education, Inc.

Copyright © 2012 Pearson Education, Inc.

9.4 Properties of Logarithmic

Functions

■ Logarithms of Products

■ Logarithms of Powers

■ Logarithms of Quotients

■ Using the Properties Together

Slide 9- 3Copyright © 2012 Pearson Education, Inc.

Logarithms of Products

The first property we discuss is related to the product rule for exponents:

.m n m na a a

Slide 9- 4Copyright © 2012 Pearson Education, Inc.

The Product Rule for LogarithmsFor any positive numbers M, N and a

(The logarithm of a product is the sum of the logarithms of the factors.)

log ( ) log log .a a aMN M N

( 1),a

Slide 9- 5Copyright © 2012 Pearson Education, Inc.

Example

Solution

that is a sum of logarithms:

log3(9 · 27) =

Express as an equivalent expressionlog3(9 · 27).

log39 + log327.

As a check, note that

log3(9 · 27) = log3243 = 5 35 = 243

and that

log39 + log327 = 2 + 3 = 5. 32 = 9 and 33 = 27

Slide 9- 6Copyright © 2012 Pearson Education, Inc.

Example

Solution

that is a single logarithm:

Express as an equivalent expression

= loga(42). Using the product

rule for logarithms

loga6 + loga7.

loga6 + loga7 = loga(6 · 7)

Slide 9- 7Copyright © 2012 Pearson Education, Inc.

Logarithms of Powers

The second basic property is related to the power rule for exponents:

.nm mna a

Slide 9- 8Copyright © 2012 Pearson Education, Inc.

The Power Rule for LogarithmsFor any positive numbers M and a and any real number p,

(The logarithm of a power of M is the exponent times the logarithm of M.)

log log .pa aM p M

( 1),a

Slide 9- 9Copyright © 2012 Pearson Education, Inc.

Example

Solution

expression that is a product:

Use the power rule to write an equivalent

= log4x1/2

Using the power rule for logarithms

a) loga6–3;

a) loga6-3 = –3loga6

4b) log .x

4b) log x

= ½ log4x Using the power rule

for logarithms

Slide 9- 10Copyright © 2012 Pearson Education, Inc.

Logarithms of Quotients

The third property that we study is similar to the quotient rule for exponents:

.m

m nn

aa

a

Slide 9- 11Copyright © 2012 Pearson Education, Inc.

The Quotient Rule for LogarithmsFor any positive numbers M, N and a

(The logarithm of a quotient is the logarithm of the dividend minus the logarithm of the divisor.)

log log log .a a aM

M NN

( 1),a

Slide 9- 12Copyright © 2012 Pearson Education, Inc.

Example

Solution

that is a difference of logarithms:

log3(9/y) =

Express as an equivalent expression

log3(9/y).

log39 – log3y. Using the quotient

rule for logarithms

Slide 9- 13Copyright © 2012 Pearson Education, Inc.

Example

Solution

that is a single logarithm:

Express as an equivalent expression

loga6 – loga7.

loga6 – loga7 = loga(6/7) Using the quotient

rule for logarithms “in reverse”

Slide 9- 14Copyright © 2012 Pearson Education, Inc.

Using the Properties Together

Slide 9- 15Copyright © 2012 Pearson Education, Inc.

Example

Solution

using individual logarithms of x, y, and z.

Express as an equivalent expression

= log4x3 – log4 yz

334 7

a) log b) logbx xy

yz z

3

4a) log x

yz= 3log4x – log4 yz

= 3log4x – (log4 y + log4z)

= 3log4x –log4 y – log4z

Slide 9- 16Copyright © 2012 Pearson Education, Inc.

1/ 33

7 7 b) log logb b

xy xy

z z

71

log3 b

xy

z

71log log

3 b bxy z

1log log 7log

3 b b bx y z

Solution continued

Slide 9- 17Copyright © 2012 Pearson Education, Inc.

CAUTION! Because the product and quotient rules replace one term with two, it is often best to use the rules within parentheses, as in the previous example.

Slide 9- 18Copyright © 2012 Pearson Education, Inc.

Example

Solution

that is a single logarithm.

Express as an equivalent expression1

log 2log log3 b b bx y z

1log 2log log

3 b b bx y z

= logbx1/3 – logb y2 + logbz1/ 3

2log logb b

xz

y

3

2logb

z x

y

Slide 9- 19Copyright © 2012 Pearson Education, Inc.

The Logarithm of the Base to an ExponentFor any base a,

(The logarithm, base a, of a to an exponent is the exponent.)

log .ka a k

Slide 9- 20Copyright © 2012 Pearson Education, Inc.

Example

Solution

Simplify: a) log668 b) log33–3.4

a) log668 =8

b) log33–3.4 = –3.4

8 is the exponent to which you raise 6 in order to get 68.

Slide 9- 21Copyright © 2012 Pearson Education, Inc.

For any positive numbers M, N, and a

( 1) :a

log ( ) log log ;a a aMN M N

log log ;pa aM p M

log log log ;a a aM

M NN

log .aka k