PP 7.4 - General Logarithmic and Exponential Functions...

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General Logarithmic and Exponential Functions
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General Exponential Functions ) ln( by defined is base ith function w l exponentia The number. real any be 0 Let a x x e a a a = a > 1 0<a<1
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Different Bases
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Laws of Exponents x x x xy y x y x y x y x y x b a ab a a a a a a a a b a y x = = = = - + ) ( . 4 ) ( . 3 . 2 . 1 then , 0 , and numbers, real are and If
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Proof of Law 4 x x b x a x b x a x b a x ab x x b a e e e e e ab = = = = = + + ) ( : get we definition the Using ) ln( ) ln( ) ln( ) ln( )] ln( ) [ln( ) ln( Students should write the proofs of the other three laws.
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Derivatives and Integrals + = = = = = = C a a du a dx du a a a dx d x u u a a a x dx d e e dx d a dx d u u u u x a x a x x ) ln( 1 and ) ln( ) ( if Hence, ) ln( )] ln( [ ) ln( ) ln(
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Examples 2 3 5 x dx d ) 5 ln( ) 6 ( 5 ) 6 )( 5 ln( 5 ) 3 ( ) 5 ln( 5 2 2 2 3 3 2 3 x x x dx d x x x = = = dx x 2 7 x u du u 2 , 7 2 1 = = C u + = 7 ) 7 ln( 1 2 1 C x + = 2 ) 7 ln( 1 2 1 7
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Comments 1 : Rule Power - = r r rx x dx d For the Power Rule to apply, x is a variable,
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PP 7.4 - General Logarithmic and Exponential Functions...

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