mac1147_lecture15_1_b

# mac1147_lecture15_1_b - L15 Properties of Logarithms...

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202 L15 Properties of Logarithms; Logarithmic and Exponential Equations; Applications Properties of Logarithms For any positive numbers x , y and for any real r , p (0 p ) the following properties hold ( 0 a > , 1 a ): log ( ) log log aa a x yx y =+ log log log a x x y y ⎛⎞ =− ⎜⎟ ⎝⎠ log log r x rx = 1 log log p a a x x p =⋅ log 1 a a = log 1 0 a = Identities ( ) log x a ax = for all real x log a x = for 0 x > Change of Base Formula If a , b , x are positive with 1 a and 1 b , then log log log b a b x x a =

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203 Example : Simplify the expressions. ln3 e = 3 ln x e = 2 4log 2 x = 4 9 9 log 3 ⎛⎞ = ⎜⎟ ⎝⎠ 64 log 2 = Example : Rewrite the expressions using properties of logarithms where it is appropriate. All variables represent positive numbers. 2 log (3 2 ) x y + x y (1 ) y 3 5 x y z =
204 Example : Use the properties of logarithms to write as a single logarithm. Find the domain. 77 7 1 2log log 3log 4 x yz +− = Common Logarithm We denote: 10 log log x x = It is called the common logarithm of x . Example : Evaluate without a calculator. log1 = log10 = log 10 = log100 = Calculators can be used to evaluate base e or base 10 logarithms. Example : log142 2.1523 ln10 2.3026

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205 Evaluate: 3 log 5 = ? Change of Base Formula If a , b , x are positive with 1 a and 1 b , then log log log b a b x x a = .
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mac1147_lecture15_1_b - L15 Properties of Logarithms...

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