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3.4
Properties of
Logarithmic
Functions
Copyright © 2011 Pearson, Inc.
What you’ll learn about



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Properties of Logarithms
Change of Base
Graphs of Logarithmic Functions with Base b
Re-expressing Data
… and why
The applications of logarithms are based on their many
special properties, so learn them well.
Copyright © 2011 Pearson, Inc.
Slide 3.4 - 2
Properties of Logarithms
Let b, R, and S be positve real numbers with b ¹ 1,
and c any real number.
i Product rule:
i Quotient rule:
i Power rule:
Copyright © 2011 Pearson, Inc.
log b (RS) = log b R + log b S
æ Rö
log b ç ÷ = log b R - log b S
è Sø
log b (R)c = c log b R
Slide 3.4 - 3
Example Proving the Product Rule
for Logarithms
Prove logb (RS) = logb R + logb S.
Copyright © 2011 Pearson, Inc.
Slide 3.4 - 4
Example Proving the Product Rule
for Logarithms
Prove logb (RS) = logb R + logb S.
Let x = log b R and y = log b S. The corresponding
exponential statements are b x = R and b y = S.
Therefore,
RS = b x × b y
RS = b x+ y
log b (RS) = x + y
change to logarithmic form
log b (RS) = log b R + log b S
Copyright © 2011 Pearson, Inc.
Slide 3.4 - 5
Example Expanding the Logarithm
of a Product
Assuming x is positive, use properties of logarithms to write
( )
log 3x 5 as a sum of logarithms or multiple logarithms.
Copyright © 2011 Pearson, Inc.
Slide 3.4 - 6
Example Expanding the Logarithm
of a Product
Assuming x is positive, use properties of logarithms to write
( )
log 3x 5 as a sum of logarithms or multiple logarithms.
( )
( )
log 3x 5 = log 3 + log x 5
= log 3 + 5 log x
Copyright © 2011 Pearson, Inc.
Slide 3.4 - 7
Example Condensing a Logarithmic
Expression
Assuming x is positive, write 3ln x - ln 2
as a single logarithm.
Copyright © 2011 Pearson, Inc.
Slide 3.4 - 8
Example Condensing a Logarithmic
Expression
Assuming x is positive, write 3ln x - ln 2
as a single logarithm.
3ln x - ln 2 = ln x - ln 2
3
æ x3 ö
= ln ç ÷
è 2ø
Copyright © 2011 Pearson, Inc.
Slide 3.4 - 9
Change-of-Base Formula for
Logarithms
For positive real numbers a, b, and
x with a ¹ 1 and b ¹ 1,
log a x
log b x =
.
log a b
Copyright © 2011 Pearson, Inc.
Slide 3.4 - 10
Example Evaluating Logarithms by
Changing the Base
Evaluate log 3 10.
Copyright © 2011 Pearson, Inc.
Slide 3.4 - 11
Example Evaluating Logarithms by
Changing the Base
Evaluate log 3 10.
log10
1
log 3 10 =
=
» 2.096
log 3 log 3
Copyright © 2011 Pearson, Inc.
Slide 3.4 - 12
Example Transforming Logarithmic
Graphs
Describe how to transform the graph of f ( x ) = ln x
into the graph of each function.
Graph each function for x > 0.
a. g(x) = ln x 2
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b. h(x) = log x
c. k(x) = ln 3x
Slide 3.4 - 13
Example Transforming Logarithmic
Graphs
Describe how to transform the graph of f ( x ) = ln x
into the graph of each function.
Graph each function for x > 0.
a. g(x) = ln x 2
Since ln x 2 = 2 ln x, the graph
of g(x) = ln x is obtained by
2
vertically stretching the graph
of f (x) = ln x by a factor of 2.
Copyright © 2011 Pearson, Inc.
Slide 3.4 - 14
Example Transforming Logarithmic
Graphs
Describe how to transform the graph of f ( x ) = ln x
into the graph of each function.
Graph each function for x > 0.
b. h(x) = log x
ln x
Since log x =
, the graph
ln10
of h(x) = log x is obtained by
vertically shrinking the graph
1
of f (x) = ln x by a factor of
» 0.43
ln10
Copyright © 2011 Pearson, Inc.
Slide 3.4 - 15
Example Transforming Logarithmic
Graphs
Describe how to transform the graph of f ( x ) = ln x
into the graph of each function.
Graph each function for x > 0.
c. k(x) = ln 3x
Since ln 3x = ln 3 + ln x, the
graph of k(x) = ln 3x is
obtained by translating the
graph of f (x) = ln x up by
ln 3 » 1.10 units.
Copyright © 2011 Pearson, Inc.
Slide 3.4 - 16
Quick Review
Evaluate the expression without using a calculator.
1. log10 3
2. ln e 3
3. log 10 -2
Simplify the expression.
x 3 y -3
4. -2 2
x y
x y )
(
5.
2
4 1/2
2x -3
Copyright © 2011 Pearson, Inc.
Slide 3.4 - 17
Quick Review Solutions
Evaluate the expression without using a calculator.
1. log10 3
2. ln e 3
3
3
3. log 10 -2
-2
Simplify the expression.
x 3 y -3
4. -2 2
x y
x y )
(
5.
2
4 1/2
2x -3
Copyright © 2011 Pearson, Inc.
x5
y5
x 4 y2
2
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