1. Section 6.2* The Natural Logarithmic Function
MAT 1226 Calculus II
Section 6.2*
The Natural Logarithmic Function

2. Homework and …
WebAssign HW 6.2* (19 problems, WA average=122 min.)

3. What’s First?

4. Preview

5. Preview
Define the Natural log. function ln⁡(𝑥) as the antiderivative of 1/𝑥
The definition gives all the properties of log. functions we learned in algebra class
Extend the antiderivative of 1/𝑥 to ln⁡|𝑥|
Log. Differentiation

6. In the beginning…
In the sequence 6.2*-6.4*, we construct the general exponential and log. function last.
For example, the function
is undefined at this point.

7. To save time... We will skip some of the obvious algebraic steps
We will not copy-n-paste everything
(Do not do these in exams/quizzes)

8. Definition
The Natural Log. Function is defined as
for 𝑥>0

9. Geometric Meaning

10. Geometric Meaning

11. Properties

12. Example 1

13. Property (3)
Suppose 𝑥,𝑦>0, and 𝑟 = rational number

14. Property (3)
Suppose 𝑥,𝑦>0, and 𝑟 = rational number

15. Property (3)
Suppose 𝑥,𝑦>0, and 𝑟 = rational number

16. Property (3)
Suppose 𝑥,𝑦>0, and 𝑟 = rational number

17. Example 2

18. Example 3

19. Example 4

20. Example 4 Direct differentiation
Take the log. first and then differentiate
(Log. Differentiation)

21. Example 4

22. Example 5

23. Example 5

24. Example 5

25. Example 5

26. Property (8)

27. Example 6

28. Example 7

29. Formula