2. 5-1: Natural Logs and Differentiation Objectives: ©2003Roy L. Gover (www.mrgover.com) Review properties of natural logarithms Differentiate natural logarithm functions

3. Review

4. Example Write as a natural logarithm:

5. Example Write in exponential form:

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

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

8. Try This Expand the log function:

9. Try This Expand the log function: then:

10. Try This Expand the log function: then:

11. Try This Evaluate:

12. Try This Evaluate:

13. Try This Evaluate: 1.6094

14. Try This Write as a logarithm of a single quantity:

15. Try This Write as a logarithm of a single quantity:

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

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

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

19. Definition from the previous definition... therefore: memorize

20. The chain rule version: Definition

21. Examples

22. Try This

23. Example Hint: use the product rule

24. Example Hint: use the quotient rule

25. Example Is this a quotient rule problem?

26. Example

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

28. Solution Rewrite:

29. Solution Use chain rule:

30. Try This Rewrite using log properties before differentiation...

31. Rewrite: Solution

32. …then differentiate Solution And simplify:

33. 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.

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

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

36. 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.

37. Assignment 1. 324/15-29 Odd (Slides 1-14) 2. 324/31-63 Odd (Slides 15-36)