Download Topic 5.3 - CSE

Topic 5.3
Properties of
Logarithms
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2012,
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Topic 5.3,
1.1, Slide 1
OBJECTIVES
1. Using the Product Rule, Quotient Rule, and
Power Rule for Logarithms
2. Expanding and Condensing Logarithmic
Expressions
3. Solving Logarithmic Equations Using the
Logarithm Property of Equality
4. Using the Change of Base Formula
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Topic 5.3,
1.1, Slide 2
Using the Product Rule, Quotient
Rule, and Power Rule for Logarithms
Properties of Logarithms
If b>0, b≠0,u and v, represent positive numbers and r is any
real number, then
Product rule for logarithms
Quotient rule for logarithms
Power rule for logarithms
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2012,
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2014 Pearson Education, Inc.
Topic 5.3,
1.1, Slide 3
Using the Product Rule, Quotient
Rule, and Power Rule for Logarithms
EXAMPLE
Use the product rule for logarithms to expand each expression.
Assume x>0.
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2012,
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2014 Pearson Education, Inc.
Topic 5.3,
1.1, Slide 4
Using the Product Rule, Quotient
Rule, and Power Rule for Logarithms
EXAMPLE
Use the quotient rule for logarithms to expand each expression.
Assume x>0.
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2012,
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2014 Pearson Education, Inc.
Topic 5.3,
1.1, Slide 5
Using the Product Rule, Quotient
Rule, and Power Rule for Logarithms
EXAMPLE
Use the power rule for logarithms to rewrite each expression. Assume
x>0.
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2012,
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Topic 5.3,
1.1, Slide 6
Expanding and Condensing
Logarithmic Expressions
EXAMPLE
Use properties of logarithms to expand each logarithmic expression as
much as possible.
Product rule for logarithms
Product rule for logarithms
Power rule for logarithms
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2012,
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Topic 5.3,
1.1, Slide 7
Expanding and Condensing
Logarithmic Expressions
EXAMPLE continued
Use properties of logarithms to expand each logarithmic expression as
much as possible.
 ( x − 2 )( x + 2 ) 
= ln 

x3
9e


Quotient rule for logarithms
ln( x − 2) + ln( x + 2) − ln 9 + ln e x 


3
Product rule for logarithms (twice)
ln( x − 2) + ln( x + 2) − ln 9 + x3 
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2012,
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2014 Pearson Education, Inc.
Topic 5.3,
1.1, Slide 8
Expanding and Condensing
Logarithmic Expressions
EXAMPLE
Use properties of logarithms to rewrite each expression as a single
logarithm.
1
a. log( x − 1) − 3 log z + log 5
2
log( x − 1)
1
( x − 1)
log
z3
2
1
5( x − 1)
log
z3
− log z + log 5
2
1
3
2
Use the power rule twice
Quotient rule for logarithms
+ log 5
Product rule for logarithms
5 x −1
or log
z3
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2012,
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Topic 5.3,
1.1, Slide 9
Expanding and Condensing
Logarithmic Expressions
EXAMPLE continued
Use properties of logarithms to rewrite
each expression as a single logarithm.
1
(log3 x − log3 y2 ) + log3 10
3
x
1
(log3 1 ) + log3 10
y
3
1

 x  3
log3  1   + log3 10
y  




  x 1 3 

x 
log3 10  2   or log3 10 3 2 
y 
 y  



1
b. (log3 x − 2 log3 y ) + log3 10
3
Use the power rule
Quotient rule for logarithms
Use the power rule
Product rule for logarithms
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2012,
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Topic 5.3,
1.1, Slide 10
Solving Logarithmic Equations
Using the Logarithm Property of Equality
Logarithm Property of Equality
If a logarithmic equation can be written in the form logbu=logbv,
then u=v. Furthermore, if u=v, then logbu=logbv.
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Topic 5.3,
1.1, Slide 11
Solving Logarithmic Equations
Using the Logarithm Property of Equality
EXAMPLE
Solve the following equations:
x = ±4
*however x ≠ −4
so, x = 4
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Topic 5.3,
1.1, Slide 12
Using the Change of Base Formula
Change of Base Formula
For any positive base b≠1 and any positive real number u,
then
Where a is any positive number such that a≠1.
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Topic 5.3,
1.1, Slide 13
Using the Change of Base Formula
EXAMPLE
Approximate the following expressions. Round each to four decimal
places.
ln π
≈ 2.0840
ln 3
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Topic 5.3,
1.1, Slide 14
Using the Change of Base Formula
EXAMPLE
Use the change of base formula and the properties of logarithms to
solve the following equation:
But x ≠ −2,
x = ±2
so, x = 2
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2012,
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Topic 5.3,
1.1, Slide 15