M 112 Short Course in Calculus

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Presentation transcript:

M 112 Short Course in Calculus Chapter 1 – Functions and Change Sections 1.6 – The Natural Logarithm V. J. Motto

1.6 The Natural Logarithm The natural logarithm of x, written, ln x, is the power of e needed to get x. In other words, ln x = c means ec = x The natural logarithm is sometimes written loge x. 8/7/2018

Illustration 1 Properties of logarithms ln e3 = 3 since 3 is the power of e needed to give e3. ln(1/e) = ln e-1 = -1 A calculator gives ln 5 = 1.6094 ln(-7) does not exist. The calculator gives us an error message. (Why?) What about ln(0)? 8/7/2018

Properties of Natural Logarithms 8/7/2018

The graph of ln x, ex , and x y = ex y = x y = ln x What do you observe about the functions? 8/7/2018

Example 1: (page 47) 8/7/2018

Example 2: (page 48) 8/7/2018

Example 3: (page 48) 8/7/2018

Exponential Functions with Base e 8/7/2018

Example 4: (page 49) 8/7/2018

Example 5: (page 49) 8/7/2018