1. § 4.6
Properties of the Natural Logarithm Function

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

3. Properties of the Natural Logarithm Function

4. 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.

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

6. Differentiating Logarithmic Expressions
CONTINUED
Distribute.
Finish differentiating.
Simplify.

7. 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.

8. 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.

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