Chapter 3
Exponential and
Logarithmic Functions
3.3 Properties of
Logarithms
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Objectives:
•
•
•
•
•
•
Use the product rule.
Use the quotient rule.
Use the power rule.
Expand logarithmic expressions.
Condense logarithmic expressions.
Use the change-of-base property.
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The Product Rule
Let b, M, and N be positive real numbers with b  1.
logb ( MN )  logb M  logb N
The logarithm of a product is the sum of the logarithms.
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Example: Using the Product Rule
Use the product rule to expand each logarithmic
expression:
log 6 (7g11)  log 6 7  log 6 11
log(100 x)  log100  log x  2  log x
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The Quotient Rule
Let b, M, and N be positive real numbers with b  1.
M

log b    log b M  log b N
N
The logarithm of a quotient is the difference of the
logarithms.
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Example: Using the Quotient Rule
Use the quotient rule to expand each logarithmic
expression:
23 

log8    log8 23  log8 x
 x
 e5 
ln    ln e5  ln11  5  ln11
 11 
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The Power Rule
Let b and M be positive real numbers with b  1, and let
p be any real number.
logb M p  p logb M
The logarithm of a number with an exponent is the
product of the exponent and the logarithm of that
number.
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Example: Using the Power Rule
Use the power rule to expand each logarithmic
expression:
log 6 3  9log 6 3
9
1
3
1
ln x  ln x  ln x
3
3
log( x  4)2  2log( x  4)
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Properties for Expanding Logarithmic Expressions
For M > 0 and N > 0:
1. logb ( MN )  logb M  logb N
Product Rule
M

2. log b    log b M  log b N
N
Quotient Rule
3. logb M p  p logb M
Power Rule
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Example: Expanding Logarithmic Expressions
Use logarithmic properties to expand the expression as
much as possible:

logb x 4 3
1
 4 13 
y  log b  x y   logb x 4  logb y 3


1
 4log b x  log b y
3

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Example: Expanding Logarithmic Expressions
Use logarithmic properties to expand the expression as
much as possible:
1


1
2
 x 
x 
3
2

log 5 

log
x

log
25
y

log

3
5
5
5
3
 25 y 
 25 y 


1
2
1
 log5 x  (log 5 5  log 5 y )  log 5 x  (log 5 52  3log 5 y)
2
1
1
 log 5 x  2log 5 5  3log 5 y  log 5 x  2  3log 5 y
2
2
2
3
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Condensing Logarithmic Expressions
For M > 0 and N > 0:
1. logb M  logb N  logb ( MN )
Product rule
M

2. log b M  log b N  log b  
N
Quotient rule
3. p logb M  logb M p
Power rule
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Example: Condensing Logarithmic Expressions
Write as a single logarithm:
log 25  log 4  log(25g4)  log100  2
7x  6
log(7 x  6)  log x  log
x
1
1
1
2
3
2ln x  ln( x  5)  ln x  ln( x  5)  ln x 2 ( x  5) 3
3
 ln x 2 3 x  5
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The Change-of-Base Property
For any logarithmic bases a and b, and any positive
number M,
log a M
log b M 
log a b
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.
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The Change-of-Base Property: Introducing Common and
Natural Logarithms
Introducing Common Logarithms
log M
log b M 
log b
Introducing Natural Logarithms
ln M
log b M 
ln b
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Example: Changing Base to Common Logarithms
Use common logarithms to evaluate
log 2506
log 7 2506 
 4.02
log 7
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Example: Changing Base to Natural Logarithms
Use natural logarithms to evaluate
ln 2506
log 7 2506 
 4.02
ln 7
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