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OBJECTIVES:

1. Use the product rule for logarithms.

2. Use the quotient rule for logarithms.

3. Use the power rule for logarithms.

4. Use the properties to write alternative forms of logarithmic expressions.

1. Use the product rule for logarithms.

2. Use the quotient rule for logarithms.

3. Use the power rule for logarithms.

4. Use the properties to write alternative forms of logarithmic expressions.

1.
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Sec 11.4 - 1

Sec 11.4 - 1

2.
Chapter 11

Inverse, Exponential and

Logarithmic

Functions

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Sec 11.4 - 2

Inverse, Exponential and

Logarithmic

Functions

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Sec 11.4 - 2

3.
11.4

Properties of Logarithms

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Sec 11.4 - 3

Properties of Logarithms

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Sec 11.4 - 3

4.
11.4 Properties of Logarithms

Objectives

1. Use the product rule for logarithms.

2. Use the quotient rule for logarithms.

3. Use the power rule for logarithms.

4. Use the properties to write alternative forms of

logarithmic expressions.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 11.4 - 4

Objectives

1. Use the product rule for logarithms.

2. Use the quotient rule for logarithms.

3. Use the power rule for logarithms.

4. Use the properties to write alternative forms of

logarithmic expressions.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 11.4 - 4

5.
11.4 Properties of Logarithms

Product Rule for Logarithms

Product Rule for Logarithms

If x, y, and b are positive real numbers, where b ≠ 1, then

logb xy = logb x + logb y.

In words, the logarithm of a product is the sum of the logarithms of the

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 11.4 - 5

Product Rule for Logarithms

Product Rule for Logarithms

If x, y, and b are positive real numbers, where b ≠ 1, then

logb xy = logb x + logb y.

In words, the logarithm of a product is the sum of the logarithms of the

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 11.4 - 5

6.
11.4 Properties of Logarithms

Note on Solving Equations

The word statement of the product rule can be restated by replacing

“logarithm” with “exponent.” The rule then becomes the familiar rule for

multiplying exponential expressions: The exponent of a product is equal

to the sum of the exponents of the factors.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 11.4 - 6

Note on Solving Equations

The word statement of the product rule can be restated by replacing

“logarithm” with “exponent.” The rule then becomes the familiar rule for

multiplying exponential expressions: The exponent of a product is equal

to the sum of the exponents of the factors.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 11.4 - 6

7.
11.4 Properties of Logarithms

EXAMPLE 1 Using the Product Rule

Use the product rule to rewrite each expression. Assume n > 0.

(a) log4 (3 · 5)

By the product rule,

log4 (3 · 5) = log4 3 + log4 5.

(b) log5 2 + log5 8 = log5 (2 · 8) = log5 16

(c) log7 (7n)

= log7 7 + log7 n

= 1 + log7 n log7 7 = 1

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 11.4 - 7

EXAMPLE 1 Using the Product Rule

Use the product rule to rewrite each expression. Assume n > 0.

(a) log4 (3 · 5)

By the product rule,

log4 (3 · 5) = log4 3 + log4 5.

(b) log5 2 + log5 8 = log5 (2 · 8) = log5 16

(c) log7 (7n)

= log7 7 + log7 n

= 1 + log7 n log7 7 = 1

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 11.4 - 7

8.
11.4 Properties of Logarithms

EXAMPLE 1 Using the Product Rule

Use the product rule to rewrite each expression. Assume n > 0.

(d) log3 n4

= log3 (n · n · n · n) n4 = n · n · n · n

= log3 n + log3 n + log3 n + log3 n Product rule

= 4 log3 n

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 11.4 - 8

EXAMPLE 1 Using the Product Rule

Use the product rule to rewrite each expression. Assume n > 0.

(d) log3 n4

= log3 (n · n · n · n) n4 = n · n · n · n

= log3 n + log3 n + log3 n + log3 n Product rule

= 4 log3 n

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 11.4 - 8

9.
11.4 Properties of Logarithms

Quotient Rule for Logarithms

Quotient Rule for Logarithms

If x, y, and b are positive real numbers, where b ≠ 1, then

x

logb y = logb x – logb y.

In words, the logarithm of a quotient is the difference between the

logarithm of the numerator and the logarithm of the denominator.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 11.4 - 9

Quotient Rule for Logarithms

Quotient Rule for Logarithms

If x, y, and b are positive real numbers, where b ≠ 1, then

x

logb y = logb x – logb y.

In words, the logarithm of a quotient is the difference between the

logarithm of the numerator and the logarithm of the denominator.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 11.4 - 9

10.
11.4 Properties of Logarithms

EXAMPLE 2 Using the Quotient Rule

Use the quotient rule to rewrite each logarithm.

(a) log5 3 = log5 3 – log5 4

4

(b) log3 7 – log3 n 7

= log3 n , n > 0

(c) log4 64 = log4 64 – log4 9

9

= 3 – log4 9 log4 64 = 3

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 11.4 - 10

EXAMPLE 2 Using the Quotient Rule

Use the quotient rule to rewrite each logarithm.

(a) log5 3 = log5 3 – log5 4

4

(b) log3 7 – log3 n 7

= log3 n , n > 0

(c) log4 64 = log4 64 – log4 9

9

= 3 – log4 9 log4 64 = 3

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 11.4 - 10

11.
11.4 Properties of Logarithms

Caution

There is no property of logarithms to rewrite the logarithm of a sum

or difference. For example, we cannot write logb (x + y) in terms of

logb x and logb y. Also,

x logb x

logb y ≠ .

logb y

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 11.4 - 11

Caution

There is no property of logarithms to rewrite the logarithm of a sum

or difference. For example, we cannot write logb (x + y) in terms of

logb x and logb y. Also,

x logb x

logb y ≠ .

logb y

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 11.4 - 11

12.
11.4 Properties of Logarithms

Power Rule for Logarithms

Power Rule for Logarithms

If x and b are positive real numbers, where b ≠ 1, and if r is any real

number, then

logb x r = r logb x.

In words, the logarithm of a number to a power equals the exponent

times the logarithm of the number.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 11.4 - 12

Power Rule for Logarithms

Power Rule for Logarithms

If x and b are positive real numbers, where b ≠ 1, and if r is any real

number, then

logb x r = r logb x.

In words, the logarithm of a number to a power equals the exponent

times the logarithm of the number.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 11.4 - 12

13.
11.4 Properties of Logarithms

EXAMPLE 3 Using the Power Rule

Use the power rule to rewrite each logarithm. Assume b > 0, x > 0, and b ≠ 1.

(a) log5 74 = 4 log5 7

(b) logb x3 = 3 logb x

(c) log4 3 x8 = log4 x8/3 Rewrite the radical expression

with a rational exponent.

8

= 3 log4 x

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 11.4 - 13

EXAMPLE 3 Using the Power Rule

Use the power rule to rewrite each logarithm. Assume b > 0, x > 0, and b ≠ 1.

(a) log5 74 = 4 log5 7

(b) logb x3 = 3 logb x

(c) log4 3 x8 = log4 x8/3 Rewrite the radical expression

with a rational exponent.

8

= 3 log4 x

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 11.4 - 13

14.
11.4 Properties of Logarithms

Special Properties

Special Properties

If b > 0 and b ≠ 1, then

b logb x = x and logb b x = x.

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Special Properties

Special Properties

If b > 0 and b ≠ 1, then

b logb x = x and logb b x = x.

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15.
11.4 Properties of Logarithms

EXAMPLE 4 Using the Special Properties

Find the value of each logarithmic expression

(a) log3 37

Since logb b x = x, log3 37 = 7

(b) log5 625 = log5 54 = 4

(c) 3 log3 8 = 8

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 11.4 - 15

EXAMPLE 4 Using the Special Properties

Find the value of each logarithmic expression

(a) log3 37

Since logb b x = x, log3 37 = 7

(b) log5 625 = log5 54 = 4

(c) 3 log3 8 = 8

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 11.4 - 15

16.
11.4 Properties of Logarithms

Properties of Logarithms

Properties of Logarithms

If x, y, and b are positive real numbers, where b ≠ 1, and r is any real

number, then

Product Rule logb xy = logb x + logb y

x

Quotient Rule logb y = logb x – logb y

Power Rule logb x r = r logb x

Special Properties b logb x = x and logb b x = x.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 11.4 - 16

Properties of Logarithms

Properties of Logarithms

If x, y, and b are positive real numbers, where b ≠ 1, and r is any real

number, then

Product Rule logb xy = logb x + logb y

x

Quotient Rule logb y = logb x – logb y

Power Rule logb x r = r logb x

Special Properties b logb x = x and logb b x = x.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 11.4 - 16

17.
11.4 Properties of Logarithms

EXAMPLE 5 Writing Logarithms in Alternative Forms

Use the properties of logarithms to rewrite each expression. Assume all

variables represent positive real numbers.

(a) log5 5x7 = log5 5 + log5 x7 Product rule

= 1 + 7 log5 x log5 5 = 1; power rule

a a 1/3

(b) log4 3 = log4

c c

1 log a

= Power rule

3 4

c

1 ( log a – log c )

= Quotient rule

3 4 4

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 11.4 - 17

EXAMPLE 5 Writing Logarithms in Alternative Forms

Use the properties of logarithms to rewrite each expression. Assume all

variables represent positive real numbers.

(a) log5 5x7 = log5 5 + log5 x7 Product rule

= 1 + 7 log5 x log5 5 = 1; power rule

a a 1/3

(b) log4 3 = log4

c c

1 log a

= Power rule

3 4

c

1 ( log a – log c )

= Quotient rule

3 4 4

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 11.4 - 17

18.
11.4 Properties of Logarithms

EXAMPLE 5 Writing Logarithms in Alternative Forms

Use the properties of logarithms to rewrite each expression. Assume all

variables represent positive real numbers.

(c) log3 w2 2 Quotient rule

mn = log3 w – log3 mn

= 2 log3 w – log3 mn Power rule

= 2 log3 w – ( log3 m + log3 n ) Product rule

= 2 log3 w – log3 m – log3 n Distributive property

Notice the careful use of parentheses in the third step. Since we are

subtracting the logarithm of a product and rewriting it as a sum of two terms,

we must place parentheses around the sum.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 11.4 - 18

EXAMPLE 5 Writing Logarithms in Alternative Forms

Use the properties of logarithms to rewrite each expression. Assume all

variables represent positive real numbers.

(c) log3 w2 2 Quotient rule

mn = log3 w – log3 mn

= 2 log3 w – log3 mn Power rule

= 2 log3 w – ( log3 m + log3 n ) Product rule

= 2 log3 w – log3 m – log3 n Distributive property

Notice the careful use of parentheses in the third step. Since we are

subtracting the logarithm of a product and rewriting it as a sum of two terms,

we must place parentheses around the sum.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 11.4 - 18

19.
11.4 Properties of Logarithms

EXAMPLE 5 Writing Logarithms in Alternative Forms

Use the properties of logarithms to rewrite each expression. Assume all

variables represent positive real numbers.

(d) logb (h – 4) + logb (h + 3) – 4 logb h

5

= logb (h – 4) + logb (h + 3) – logb h4/5 Power rule

(h – 4)(h + 3)

= logb Product and

h4/5 quotient rules

= logb h2 – h – 12

h4/5

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 11.4 - 19

EXAMPLE 5 Writing Logarithms in Alternative Forms

Use the properties of logarithms to rewrite each expression. Assume all

variables represent positive real numbers.

(d) logb (h – 4) + logb (h + 3) – 4 logb h

5

= logb (h – 4) + logb (h + 3) – logb h4/5 Power rule

(h – 4)(h + 3)

= logb Product and

h4/5 quotient rules

= logb h2 – h – 12

h4/5

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 11.4 - 19

20.
11.4 Properties of Logarithms

EXAMPLE 5 Writing Logarithms in Alternative Forms

Use the properties of logarithms to rewrite each expression. Assume all

variables represent positive real numbers.

(e) logb (3p – 4q)

logb (3p – 4q) cannot be rewritten using the properties of logarithms.

There is no property of logarithms to rewrite the logarithm of a difference.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 11.4 - 20

EXAMPLE 5 Writing Logarithms in Alternative Forms

Use the properties of logarithms to rewrite each expression. Assume all

variables represent positive real numbers.

(e) logb (3p – 4q)

logb (3p – 4q) cannot be rewritten using the properties of logarithms.

There is no property of logarithms to rewrite the logarithm of a difference.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 11.4 - 20