§ 4.6 Properties of the Natural Logarithm Function.

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Presentation transcript:

§ 4.6 Properties of the Natural Logarithm Function

Section Outline Properties of the Natural Logarithm Function Simplifying Logarithmic Expressions Differentiating Logarithmic Expressions Logarithmic Differentiation

Properties of the Natural Logarithm Function

Simplifying Logarithmic Expressions EXAMPLE Write as a single logarithm. SOLUTION This is the given expression. Use LIV (this must be done first). Use LIII. Use LI. Simplify.

Differentiating Logarithmic Expressions EXAMPLE Differentiate. SOLUTION This is the given expression. Rewrite using LIII. Rewrite using LI. Rewrite using LIV. Differentiate.

Differentiating Logarithmic Expressions CONTINUED Distribute. Finish differentiating. Simplify.

Logarithmic Differentiation 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 factor, and finally solving for the desired derivative. Example will follow.

Logarithmic Differentiation EXAMPLE Use logarithmic differentiation to differentiate the function. SOLUTION This is the given function. Take the natural logarithm of both sides of the equation. Use LIII. Use LI.

Logarithmic Differentiation CONTINUED Use LIV. Differentiate. Solve for f ΄(x). Substitute for f (x).