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Sect+4.4-4.5+Guided+Notes

# Sect+4.4-4.5+Guided+Notes - M M a a b log log log = Now we...

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Math 1111. Guided Notes. Sections 4.4, 4.5. 4.4 Properties of Logarithmic Expressions and Functions Recall n m n m a a a + = , n m n m a a a - = , and ( 29 mn n m a a = Product Rule for Logarithms For all positive numbers M and N , and base a, N M MN a a a log log log + = Examples: 1. Expand x 5 log 3 2. Condense 15 log 4 log 5 5 + Why does the Product Rule work? Quotient Rule for Logarithms For all positive numbers M and N , and base a, N M N M a a a log log log - = Examples: 1. Expand 2 3 log 2. Condense 13 ln 6 ln - 3. Expand y x 2 2 log log 4. Condense 7 log 4 log 2 5 - Similar Proof to the Product Rule except use n m n m a a a - = .

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Power Rule for Logarithms M p M a p a log log = Examples: 1. Expand 9 5 log x 2. Condense x 2 log 3 4 Why does the Power Rule work? SPECIAL NOTE: See page 394 for Common Errors, and avoid them!! Additional Examples Expand. 1. 6 log 2 7 x 2. 2 ln z x y Condense. 3. x 2 2 log 5 15 log - 4. + - z x y log 5 log 3 2 log Other Properties of Logarithms x a x a = log x a x a = log Examples: Evaluate 1. = 5 log 10 2. = 16 ln e
Example: Given that 301 . 2 log a and 477 . 3 log a , evaluate the following, if possible. 1. 12 log a 2. 5 log a 3. 9 1 log a Change-of-Base Derivation Recall from last section: b

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Unformatted text preview: M M a a b log log log = . Now, we have the tools to verify. 4.5 Solving Exponential and Logarithmic Equations Exponential Equations – Equations with variables in the exponents. Ex. Solve 16 2 = x Logarithmic Equations – Equations that contain variables in the log expression Ex. Solve 3 log 2 = x Base-Exponent Property If y x a a = , then y x = . Logarithmic Equality Property If N M a a log log = , then N M = . Special Note: Remember that for any logarithmic expression, M a log , M is a positive number! Solving Exponential Equations Examples: 1. 32 2 3 = x 2. 33 3 1 = + x 3. 1200 = t e 4. x x 3 2 1 = + Solving Logarithmic Equation Examples: Note: Make sure that your answers do not make the “of” part of the logarithm negative!!! 1. 3 log 2-= x 2. 2 ln-= x 3. ( 29 ( 29 3 1 log 1 log 2 2 =-+ + x x 4. ( 29 ( 29 x x x ln 2 1 ln 8 ln =-+ +...
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Sect+4.4-4.5+Guided+Notes - M M a a b log log log = Now we...

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