# PP 7.2 - The Natural Logarithmic Function Definition The...

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The Natural Logarithmic Function

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Definition The natural logarithmic function is the function defined by 0 , 1 ) ln( 1 = x dt t x x This function is well defined since it is the integral of a continuous function for x > 0.
Interpretation ln(4) is the area of the region above the x-axis between the green lines, so it is a positive number. = 4 1 1 ) 4 ln( dt t

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Interpretation - = = 1 4 . 0 4 . 0 1 1 1 ) 4 . 0 ln( dt t dt t ln(0.4) is the negative of area of the region above the x-axis between the green lines, so it is a negative number.
Properties of ln(x) ln(x) > 0 if x > 1 ln(x) < 0 if 0 < x < 1 ln(1) = 0 by properties of the definite integral Domain = (0, ∞) Range = (-∞, ∞)

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Derivative of ln(x) ) ( ) ( : 1 Part Calculus, of Theorem l Fundamenta x f dt t f dx d x a = 0 1 1 ) ln( Hence, 1 = = x dt t dx d x dx d x Hence ln(x) is an increasing function.
Laws of Logarithms ) ln( ) ln( ) ln( . 1 y x xy + = ) ln( ) ln( ln . 2 y x y x - = ) ln( ) ln( . 3 x r x r =

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Proof of Law 1 Let f(x) = ln(ax), a
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## This note was uploaded on 01/21/2011 for the course PHYS 4A 60865 taught by Professor L. oldewurtel during the Fall '09 term at Irvine Valley College.

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PP 7.2 - The Natural Logarithmic Function Definition The...

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