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The Natural Logarithmic Function and Differentiation

# The Natural Logarithmic Function and Differentiation -...

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The Natural Logarithmic Function and Differentiation Section 5.1

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Defn. of the Natural Logarithmic Function 0 , 1 1 x dt t x From the defn., you can see that ln x Is positive for x>1 and negative for 0<x<1. ( Look at Figure 5.1 pg. 311.
Properties of the Natural Logarithmic Function 1. The domain is (0, ) and the range is (0, ). 2. The function is continuous, increasing, and one-to-one. 3. The graph is concave downward.

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1 2 3 1 y = ln x
Logarithmic Properties 1. ln (1) = 0 2. ln (ab) = ln a + ln b 3. ln (a n ) = n ln a 4. ln = ln a – ln b b a

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Definition of

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Unformatted text preview: denotes the positvie real number such that ln e = = 1 dt t e âˆ« 1 1 e 2.71828182846 â‰ˆ dt t e âˆ« 1 1 Area = =1 t y 1 = 1 2 3 e is the base for the natural logarithm Because ln e = 1. Derivative of ln function 1. [ ] , 1 = x x x ln dx d [ ] , ' 1 = = u u u dx du u u ln dx d 2. 3. [ ] , ' â‰  = u u u u ln dx d Joke Time What do you call a deer with no eyes? I-have-no-eye-deer (I have no idea!) What did the teddy bear say after eating a whole bag of cookies? Iâ€™m stuffed!...
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