Chapter 4 Exponential and Logarithmic Functions Copyright © 2014, 2010, 2007 Pearson Education,...

Chapter 4Exponential and Logarithmic Functions

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1

4.3 Properties of Logarithms


• Use the product rule.• Use the quotient rule.• Use the power rule.• Expand logarithmic expressions.• Condense logarithmic expressions.• Use the change-of-base property.

Objectives:


The Product Rule

Let b, M, and N be positive real numbers with b 1.

The logarithm of a product is the sum of the logarithms.

log ( ) log logb b bMN M N

Think exponent rules….

xa * xb = x a +b


Example: Using the Product Rule

Use the product rule to expand each logarithmic expression:

6log (7 11) 6 6log 7 log 11

log(100 )x log100 log x 2 log x



The Quotient Rule

Let b, M, and N be positive real numbers with b 1.

The logarithm of a quotient is the difference of the logarithms.

log log logb b bM M NN

Think exponent rules….

xa / xb = x a - b


Example: Using the Quotient Rule

Use the quotient rule to expand each logarithmic expression:

823logx

8 8log 23 log x

5

ln11e

5ln ln11e 5 ln11


The Power Rule

Let b and M be positive real numbers with b 1, and let p be any real number.

The logarithm of a number with an exponent is the product of the exponent and the logarithm of that number.

log logpb bM p M


Example: Using the Power Rule

Use the power rule to expand each logarithmic expression:

96log 3 69log 3

3ln x13ln x

1ln3

x

2log( 4)x 2log( 4)x


Properties for Expanding Logarithmic Expressions

For M > 0 and N > 0:

1. Product Rule

2. Quotient Rule

3. Power Rule


log log logb b bM M NN

log logpb bM p M


Example: Expanding Logarithmic Expressions

Use logarithmic properties to expand the expression as much as possible:

4 3logb x y1

4 3log logb bx y

14log log3b bx y

14 3logb x y


Example: Expanding Logarithmic Expressions

Use logarithmic properties to expand the expression as much as possible:

5 3log25

xy

12

5 3log25

xy

1

325 5log log 25x y

12 32

5 5 5log (log 5 log )x y 25 5 5

1 log (log 5 3log )2

x y

5 5 51 log 2log 5 3log2

x y 5 51 log 2 3log2

x y


Condensing Logarithmic Expressions

For M > 0 and N > 0:

1. Product rule

2. Quotient rule

3. Power rule

log log log ( )b b bM N MN

log log logb b bMM NN

log log pb bp M M


Example: Condensing Logarithmic Expressions

Write as a single logarithm:

log 25 log 4 log(25 4) log100 2

log(7 6) logx x 7 6log xx

12ln ln( 5)3

x x 1

2 3ln ln( 5)x x 1

2 3ln ( 5)x x 2 3ln 5x x


The Change-of-Base Property

For any logarithmic bases a and b, and any positive number M,

The logarithm of M with base b is equal to the logarithm of M with any new base divided by the logarithm of b with that new base.

logloglog

ab

a

MMb


The Change-of-Base Property: Introducing Common and Natural Logarithms

Introducing Common Logarithms

Introducing Natural Logarithms

logloglogb

MMb

lnloglnb

MMb


Example: Changing Base to Common Logarithms

Use common logarithms to evaluate

7log 2506 log 2506log7

4.02


Example: Changing Base to Natural Logarithms

Use natural logarithms to evaluate

7log 2506 ln 2506ln 7

4.02

Chapter 4 Exponential and Logarithmic Functions Copyright © 2014, 2010, 2007 Pearson Education,...

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