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Copyright © 2014, 2010, 2007 Pearson Education, Inc. 3 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. Think exponent rules…. x a * x b = x a +b
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Chapter 4Exponential and Logarithmic Functions
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1
4.3 Properties of Logarithms
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 2
• 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:
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 3
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
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 4
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
log ( ) log logb b bMN M N
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 5
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
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 6
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
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 7
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
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 8
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
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 9
Properties for Expanding Logarithmic Expressions
For M > 0 and N > 0:
1. Product Rule
2. Quotient Rule
3. Power Rule
log ( ) log logb b bMN M N
log log logb b bM M NN
log logpb bM p M
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 10
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
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 11
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
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 12
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
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 13
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
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 14
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
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 15
The Change-of-Base Property: Introducing Common and Natural Logarithms
Introducing Common Logarithms
Introducing Natural Logarithms
logloglogb
MMb
lnloglnb
MMb
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 16
Example: Changing Base to Common Logarithms
Use common logarithms to evaluate
7log 2506 log 2506log7
4.02
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 17
Example: Changing Base to Natural Logarithms
Use natural logarithms to evaluate
7log 2506 ln 2506ln 7
4.02