Section 6.2* The Natural Logarithmic Function MAT 1226 Calculus II Section 6.2* The Natural Logarithmic Function http://myhome.spu.edu/lauw

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

What’s First?

Preview

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

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.

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)

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

Geometric Meaning

Geometric Meaning

Properties

Example 1

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

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

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

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

Example 2

Example 3

Example 4

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

Example 4

Example 5

Example 5

Example 5

Example 5

Property (8)

Example 6

Example 7

Formula