5-1: Natural Logs and Differentiation Objectives: ©2003Roy L. Gover ( Review properties of natural logarithms Differentiate natural logarithm functions

Review

Example Write as a natural logarithm:

Example Write in exponential form:

Properties of Logarithms ln(1)=0 ln(ab)=ln a + ln b ln a n =n ln a ln e=1

What is the domain? What is the range? Important Idea Wassup at x =1? The graph of the natural log function looks like:

Try This Expand the log function:

Try This Expand the log function: then:

Try This Expand the log function: then:

Try This Evaluate:

Try This Evaluate:

Try This Evaluate:

Try This Write as a logarithm of a single quantity:

Try This Write as a logarithm of a single quantity:

Try This Find the antiderivative: Can’t solve using the power rule

Important Idea There exists an area under the curve equal to 1 is an area under

Definition 1 e is the positive real number such that: Area = 1

Definition from the previous definition... therefore: memorize

The chain rule version: Definition

Examples

Try This

Example Hint: use the product rule

Example Hint: use the quotient rule

Example Is this a quotient rule problem?

Example

Try This Find the derivative: Hint: Rewrite using log properties then use chain rule

Solution Rewrite:

Solution Use chain rule:

Try This Rewrite using log properties before differentiation...

Rewrite: Solution

…then differentiate Solution And simplify:

Definition Since ln x is not defined for negative values of x, you may frequently see ln| x |. The absolute value rule for ln is: When differentiating a logarithm, you may ignore any absolute value sign.

Try This Find the derivative: Don’t forget the chain rule

Try This Find the equation of the line tangent to: at (1,1)

Lesson Close The natural logarithm is frequently used in Calculus. Be certain that you understand the properties of logarithms and know how to differentiate and integrate (next section) logarithmic functions.

Assignment /15-29 Odd (Slides 1-14) /31-63 Odd (Slides 15-36)