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# 1 lim x a x 0 lim x a x if 0 a 1 8 the graph of 2 x

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1 lim x →∞ a x = 0, lim x →-∞ a x = if 0 < a < 1 8) The graph of 2 x is given below. The graph of a x , for any a > 1, is similar. x y 1 2 3 - 1 - 2 - 3 1 2 4 6 y = 2 x

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Properties of Logarithms In the following, x and y are arbitrary real numbers that are strictly bigger than 0, a is an arbitrary constant that is strictly bigger than one and e is 2.7182818284, to ten decimal places. 1) e ln x = x , a log a x = x , log e x = ln x , log a x = ln x ln a 2) log a ( a x ) = x , ln ( e x ) = x ln 1 = 0, log a 1 = 0 ln e = 1, log a a = 1 3) ln( xy ) = ln x + ln y , log a ( xy ) = log a x + log a y 4) ln ( x y ) = ln x - ln y , log a ( x y ) = log a x - log a y ln ( 1 y ) = - ln y , log a ( 1 y ) = - log a y , 5) ln( x y ) = y ln x , log a ( x y ) = y log a x 6) d dx ln x = 1 x , d dx ln( g ( x )) = g 0 ( x ) g ( x ) , d dx log a x = 1 x ln a 7) R 1 x dx = ln | x | + C , R ln x dx = x ln x - x + C 8) lim x →∞ ln x = , lim x 0 ln x = -∞ lim x →∞ log a x = , lim x 0 log a x = -∞ 9) The graph of ln x is given below. The graph of log a x , for any a > 1, is similar. x y 1 2 3 4 0 . 5 1 . 0 1 . 5 - 0 . 5 - 1 . 0 - 1 . 5 y = ln x
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• Spring '03
• Unknown
• Derivative, Inverse trigonometric functions, ln x ln

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