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lecture25

# lecture25 - Math 121 Lecture 25 4.6 Properties of the...

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§ 4.6 Properties of the Natural Logarithm Function Math 121 Lecture 25 ! Properties of the Natural Logarithm Function ! Simplifying Logarithmic Expressions ! Differentiating Logarithmic Expressions ! Logarithmic Differentiation

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Write as a single logarithm. ln2 " ln5 + 2ln4 This is the given expression. ln2 " ln5 + 2ln4 Use LIV (this must be done first). ln2 " ln5 + ln4 2 Use LI&LIII. ln 2 " 4 2 5 Simplify. ln 32 5
Write as a single logarithm. 5ln x " 1 2 ln y + 3ln z This is the given expression. Use LIV (this must be done first). ln x 5 " ln y 1 2 + ln z 3 Use LIII. ln x 5 y 1 2 + ln z 3 Use LI. ln x 5 y 1 2 z 3 " # \$ % & Simplify. Differentiate. ln x x + 1 ( ) 2 x + 2 ( ) 3 4 x + 1 " # \$ \$ % & This is the given expression. ln x x + 1 ( ) 2 x + 2 ( ) 3 4 x + 1 " # \$ \$ % & Rewrite using LIII. ln x x + 1 ( ) 2 x + 2 ( ) 3 [ ] " ln 4 x + 1 ( ) Rewrite using LI. ln x + ln x + 1 ( ) 2 + ln x + 2 ( ) 3 " ln 4 x + 1 ( ) Rewrite using LIV. 1 2 ln x + 2ln x + 1 ( ) + 3ln x + 2 ( ) " ln 4 x + 1 ( ) Differentiate. d dx 1 2 ln x + 2ln x + 1 ( ) + 3ln x + 2 ( ) " ln 4 x + 1 ( ) # \$ % & (

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Distribute. Finish differentiating. Simplify. Definition Example Logarithmic Differentiation : Given a function y = f ( x ), take the natural logarithm of both sides of the equation, use logarithmic rules to break up the right side of the equation into any number of factors, differentiate each
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lecture25 - Math 121 Lecture 25 4.6 Properties of the...

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