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

2. 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.
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3. 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 =
ln(-7) does not exist. The calculator gives us an error message. (Why?)
What about ln(0)?
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4. Properties of Natural Logarithms
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5. The graph of ln x, ex , and x y = ex y = x y = ln x
What do you observe about the functions?
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6. Example 1: (page 47)
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7. Example 2: (page 48)
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8. Example 3: (page 48)
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9. Exponential Functions with Base e
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10. Example 4: (page 49)
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11. Example 5: (page 49)
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