section 4.4 laws of logarithms
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Chapter 4 – Exponential and Logarithmic Functions. Section 4.4 Laws of Logarithms. Example. Find the pattern. The Product Rule. Let a be a positive number with a 1. Let A , B , and C be any real number with A > 0 and B > 0. - PowerPoint PPT PresentationTRANSCRIPT
4.4 - Laws of Logarithms
Section 4.4 Laws of
Logarithms
Chapter 4 – Exponential and Logarithmic Functions
4.4 - Laws of Logarithms
ExampleFind the pattern.
2
log[( 2)(3 7)] log( 2) log(3 7)
ln(2 ) ln(2) ln( )
ln( 3) ln( 1) ln[( 3)( 1)] ln( 2 3)
x x x x
x x
x x x x x x
4.4 - Laws of Logarithms
The Product RuleLet a be a positive number with a 1. Let A, B, and C be any real number with A > 0 and B > 0.
This means, the logarithm of a product of numbers is the sum of logarithms of the numbers.
log log ( ) log ( )a a aAB A B
4.4 - Laws of Logarithms
ExampleFind the pattern.
22
2ln ln( 2) ln( 7)
7
8log log(8) log( )
3 4 ( 4)( 1)log( 3 4) log( 1) log log log( 4)
1 1
xx x
x
xx
x x x xx x x x
x x
4.4 - Laws of Logarithms
The Quotient RuleLet a be a positive number with a 1. Let A, B, and C be any real number with A > 0 and B > 0.
This means, the logarithm of a quotient of numbers is the difference of logarithms of the numbers.
log log ( ) log ( )a a a
AA B
B
4.4 - Laws of Logarithms
ExampleFind the pattern.
7
13
2 2 3
log(( 2) ) 7 log( 2)
ln( ) 13ln( )
3ln( 7 9) ln( 7 9)
x x
x x
x x x x
4.4 - Laws of Logarithms
The Power RuleLet a be a positive number with a 1. Let A, B, and C be any real number with A > 0 and B > 0.
This means, the logarithm of a power of a number is the exponent times the logarithm of the number.
log log ( )Ca aA C A
4.4 - Laws of Logarithms
Change of Base
log lnlog
log lnb
x xx or
b b
4.4 - Laws of Logarithms
Compress the Expression
Compress the expression to a single logarithm.
19ln( 1) ln(2 4)
2x x
4.4 - Laws of Logarithms
Expand the Expression
Expand.
3
1 2 3log
2
x x
x
4.4 - Laws of Logarithms
More Examples – pg. 329
Evaluate the expression.
12 12
2 2 2
332
12. log 9 log 16
13. log 6 log 15 log 20
16. log 8
4.4 - Laws of Logarithms
More Examples – pg. 329
Use the Law of Logarithms to expand the expression.
2
22 3
2 4
37. ln
441. log
1 7
1044. log
1 2
x
yxz
x
x x
x x x
4.4 - Laws of Logarithms
More Examples – pg. 329
Use the Laws of Logarithms to combine the expression.
2 2 2
2
23 4 2
47. log log 2log
51. ln 5 2ln 3ln 5
1 153. log 2 log log 6
3 2
A B C
x x
x x x x
4.4 - Laws of Logarithms
More Examples – pg. 330
Use the Change of Base Formula and a calculator to evaluate the logarithm, rounded to 6 decimal places. Use either natural or common logarithms.
6
12
60. log 532
62. log 2.5