mac1140_lecture24_2 - L24 5.4 Evaluating Logs and Changing...

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210 L24 §5.4 Evaluating Logs and Changing Base Calculators can be used to evaluate base e or base 10 logarithms. Notation : 10 log log x x = Example : We use a calculator to find log142 2.1523 ln10 2.3026 Example : Calculate 3 log 5 . Let's change log ln a x x . log a y x = Thus, ln log ln a x x a = . So, 3 ln5 log 5 1.4650 ln3 = 211 Change-of-Base Theorem: For any positive real numbers , , and x a b , where 1 and 1 a b : log log log b a b x x a = . Example : Compute: = 20 log 6 = 5 log 16
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212 §5.5 Exponential and Logarithmic Equations Recall the procedure of solving an exponential equation: 2 3 5 32 x + = If it is possible , we write each side as a power of the same base and use the property of exponents: x y a a x y = = ( 0, 1) a a > The more general method of solving exponential equations is the following: To solve an exponential equation: 1. Reduce an equation to one of the forms ( ) f x a b = , ( ) ( ) f x g x a a = , ( ) ( ) f x g x a b = 2. Compose logarithmic function to both sides (it brings down the exponents with the variable). 3. Simplify and solve for the variable. 213 Example . Solve each equation: a) 500 300 x e = b) 2 5 2 t =
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214 c) 5 2 2 8 a e + = d) 4 0.08 100 1 221 4 t
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