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Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.6, Slide 1
Chapter 5Logarithmic Functions
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.6, Slide 2
5.6 More Properties of Logarithms
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.6, Slide 3
Product Property for Logarithms
For x > 0, y > 0, b > 0, and b ≠ 1,
logb(x) + logb(y) = logb(xy)
In words, the sum of logarithms is the logarithm of the product of their inputs.
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.6, Slide 4
Example: Using the Product Property for Logarithms
Simplify. Write the sum of logarithms as a single logarithm.
1. logb(2x) + logb(x) 2. 3 logb(x2) + 2 logb(6x)
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.6, Slide 5
Solution
log 2 log ( ) log (2 )b b bx x x x
2log 2b x
2 32 2log log (6 ) lo3 2 g log 6b b b bx x x x
3 22log 6b x x 6 2log 36b x x
8log 36b x
1.
2.
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.6, Slide 6
Applying Properties
Warning
To apply the product property for logarithms, the coefficient of each logarithm must be 1. You may need to apply the power property first!
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.6, Slide 7
Quotient Property for Logarithms
For x > 0, y > 0, b > 0, and b ≠ 1,
In words, the difference of two logarithms is the logarithm of the quotient of their inputs.
log ( ) log ( ) logb b b
xx y
y
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.6, Slide 8
Example: Product and Quotient Properties
Simplify. Write the result as a single logarithm with a coefficient of 1.
1. 1ogb(6w6) – logb(w2)
2. 2 logb(3p) + 3 logb(p2) – 4 logb(2p)
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.6, Slide 9
Solution
1. 7 27
2
6log 6 log logb b bw
ww
w
5log 6b w
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.6, Slide 10
Solution
2. 2log 3 log2 3 4log 2b b bp p p
32 42log 3 log log 2b b bp p p
32 42log 3 log 2b bp p p
32 2
4
2
4
6
3log
2
9log
16
b
b
p
p
pp
p
p
4
8
4
9log
16
9log
16
b
b
p
p
p
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.6, Slide 11
Example: Solving a Logarithmic Equation
Solve 5 52log 3 4log 2 3.x x
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.6, Slide 12
Solution 5 52 4log 3 log 2 3x x
5
2 4
5log 3 log 2 3x x
2 4
5log 3 2 3x x 5
2 4log 9 16 3x x 6
5log 144 3x 3 65 144x6 125
144x
1 6125
0.9767144
x
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.6, Slide 13
Change-of-Base Property
For a > 0, b > 0, a ≠ 1, b ≠ 1, and x > 0,
log ( )log ( )
log ( )a
ba
xx
b
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.6, Slide 14
Change-of-Base
To find a logarithm to a base other than 10, we use the change-of-base property to convert to log10; then we can use the log key on a calculator.
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.6, Slide 15
Example: Converting to log10
Find log2(12).
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.6, Slide 16
Solution
Using the log key on a calculator, we compute
So, log2(12) ≈ 3.5850.
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.6, Slide 17
Example: Using the Change-of-Base Property
Write as a single logarithm.7
7
log ( )log (4)
x
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.6, Slide 18
Solution
74
7log ( )log ( )
log (4)x
x
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.6, Slide 19
Example: Using a Graphing Calculator to Graph a Logarithmic Function
Using a graphing calculator to draw the graph of y = log3(x).
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.6, Slide 20
Solution
Use the change-of-base property,
Using the log key on the graphing calculator, enter the function and draw the graph.
3
log(3
)log ( ) .
log( )x
x
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.6, Slide 21
Comparing Properties of Logarithms
Warning
It is common to confuse the quotient property and the change-of-base property for logarithms. In general,
and
loglog ( ) log ( )
log ( )b
b bb
xx y
y
log ( )log
log ( )b
bb
xxy y
