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Bases Other than e

# Bases Other than e - a differentiable function of x 1-= n n...

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Bases Other than e Section 5.5

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Defn. of Exponential Function to Base a a x = e (ln a)x , a= 1, a >0 If a = 1, then y = 1 x = 1 is a constant function.
a 0 = 1 a x a y = a x+y a x = a x-y a y (a x ) y = a xy

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Defn. of Logarithmic Function to Base a If a >0 ,a = 1 and x >0, then the logarithmic function to the base a is denoted by log a x and is defined as x a x a ln ln 1 log =
y x xy a a a log log log + = y x y x a a a log log log - = 0 1 log = a log log = xy a y x xy a a a log log log + = x n x a n a log log =

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Properties of Inverse Functions y x iff a y a x log = = 0 , log = x for x a x a x all for x a x a , log =
The logarithmic function to the base 10 is called the common logarithmic function . y = 10 x iff x = log 10 y

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Derivatives for Bases Other than e [ ] x x a a a dx d ) (ln = [ ] dx du a a a dx d u u ) (ln = [ ] x a x dx d a ) (ln 1 log =
[ ] dx du u a u dx d a ) (ln 1 log =

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C a a dx a x x + = ln 1
Thm 5.14 The Power Rule for Real Exponents Let n be any real number and let u be a differentiable function of

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Unformatted text preview: a differentiable function of x . [ ] 1-= n n nx x dx d [ ] dx du nu u dx d n n 1-= A Limit Involving e e x x x x x x x = + = + ∞ → ∞ → 1 1 1 lim lim Compound Interest Formulas Let P = amt. of deposit, t = # of yrs ., A = balance after t yrs., r = annual interest rate, and n = # of compoundings per yr. 1. Compounding n times per yr.: nt n r P A + = 1 2. Compounded continuously: rt Pe A = Joke Time What happened to the trigonometry teacher with amnesia? He lost his identities! What math is discussed between seabirds? Intergull calculus...
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