1. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 1 of 55 § 4.6 Properties of the Natural Logarithm Function

2. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 2 of 55  Properties of the Natural Logarithm Function  Simplifying Logarithmic Expressions  Differentiating Logarithmic Expressions  Logarithmic Differentiation Section Outline

3. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 3 of 55 Properties of the Natural Logarithm Function

4. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 4 of 55 Simplifying Logarithmic ExpressionsEXAMPLE SOLUTION Write as a single logarithm. This is the given expression. Use LIV (this must be done first). Use LIII. Use LI. Simplify.

5. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 5 of 55 Differentiating Logarithmic ExpressionsEXAMPLE SOLUTION Differentiate. This is the given expression. Rewrite using LIII. Rewrite using LI. Rewrite using LIV. Differentiate.

6. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 6 of 55 Differentiating Logarithmic Expressions Distribute. CONTINUED Finish differentiating. Simplify.

7. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 7 of 55 Logarithmic Differentiation DefinitionExample 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.

8. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 8 of 55 Logarithmic DifferentiationEXAMPLE SOLUTION Use logarithmic differentiation to differentiate the function. This is the given function. Take the natural logarithm of both sides of the equation. Use LIII. Use LI.

9. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 9 of 55 Logarithmic Differentiation Use LIV. CONTINUED Differentiate. Solve for f ΄(x). Substitute for f (x).