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PP Section 5.4

# PP Section 5.4 - 2 x x = 3 log 2 2 1 x x 3 log log 2 2 2...

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Properties of Logarithms

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log a a = 1 a M a M log = r a r a = log 0. a number real any for fact that the from follows This 1 = a a These two properties follow from the fact that the logarithmic function to the base a is the inverse of the exponential function to the base a . Properties of Logarithms
( 29 log log log a a a MN M N = + y x y x a a a + = fact that the from follows This log log log a a a M N M N  = - y x y x a a a - = fact that the from follows This log log a a N N 1  = - y y a a - = 1 fact that the from follows This

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log log a r a M r M = ( 29 rx r a = x a fact that the from follows This All the properties of logarithms follow from the properties of exponents since x a x y = = a log y
Examples  Evaluate each expression. 0 1 log 32=5 OR:  log (4)+ log (8)=2+3=5 8 2 log 2 2 ) 2 ( log 4 1 log 2 2 2 - = = - 2 3 1 ) 8 ( log ) 2 ( log : OR 2 2 - = - = -

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= log 10  10 2  =2 ) 7 ( log 5 5 7 ) 81 ( log 3 4 ) 3 ( log 4 3 = 9 1 log 3 2 ) 9 ( log 3 - = - 5 3 4 log 4 log 5 3
Examples Rewrite each expression using the properties of logarithms. + 1 3 log 2 5 x x ) 1 ( log ) 3 ( log 2 5 5 + - = x x ) 1 ( log ) ( log ) 3 ( log 2 5 5 5 + - + = x x 3 log 2 + x

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Unformatted text preview: 2 + x x + = 3 log 2 2 1 x x [ ] ) 3 ( log ) ( log 2 2 2 1 +-= x x Examples Rewrite each expression using a single logarithm. ) 1 log( ) 1 log( ) log( 2 2 +-+ + x x x + + 1 ) 1 ( log 2 2 x x x [ ] ) ln( 2 ) 1 ln( ) 3 ln( 5 x x x--+ + 5 2 ) 1 )( 3 ( ln -+ x x x Change of Base Formula Calculators have keys to compute base 10 (log) and base e (ln) logarithms. Some times it is necessary to compute the logarithm of a number using another base. log log log a b b M M a = = log log M a = ln ln M a 63 log Calculate 5 log log log 5 63 63 5 = = ln ln 63 5 ≈ 2 574 . Examples 1495 log Calculate 2 2 log 1495 log 1495 log 2 = 2 ln 1495 ln = 5459 . 10 ≈...
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PP Section 5.4 - 2 x x = 3 log 2 2 1 x x 3 log log 2 2 2...

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