MTH 252 Integral Calculus Chapter 6 Integration Section 6 9 Logarithmic Functions from the Integral Point of View Copyright 2005 by Ron Wallace all rights reserved The Area Function 1 A x dt 1 t x A x 0 A x 0 1 x Domain x 0 A 1 0 1 Derivative A x x FTC Pt II x Range 1 A x What function has these characteristics A x ln x x1 The Natural ln x dt x 0 1 t Logarithm Function Properties of Logarithms ln ab ln a ln b Proof d 1 1 ln ax a dx ax x ln ax ln x k Functions w same derivative differ by a constant If x 1 ln a ln1 k k ln a ln ax ln x ln a ln ab ln a ln b Let x b x1 The Natural ln x dt x 0 1 t Logarithm Function Properties of Logarithms ln ab ln a ln b ln 1 b ln b Let a 1 b in above ln a b ln a ln b a b a 1 b use above ln a r r ln a 3 Cases r pos integer r neg integer r rational The Natural Exponential Function y e x is the function where x ln y d x 1 e d dx ln x x dx e 1 1 x ex x ex Derivative of inverse rule ex
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