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Logarithmic Functions Section 8.4

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WHAT YOU WILL LEARN: 1.How to evaluate logarithmic functions.

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Question? What is 2 2 ? What is 2 3 ? For what value of x is 2 x = 6? To solve problems like this, mathematicians developed the concept of logarithms.

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Definition of Logarithm Let b and y be positive numbers, with b not equal 1. The logarithm of y with base b is denoted by log b y = x and is defined as follows: log b y = x if and only if b x = y We read the expression as “ log base by of y ”.

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Examples Rewrite the following in exponential form: b x = y 1. log 2 32 = 5 2. log 5 1 = 0 3. log 10 10 = 1 4. log 10 0.1 = -1

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Special Logarithms Logarithm of 1log b 1 = 0 because b 0 = 1 Log of base blog b b = 1 because b 1 = b

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Evaluating Logarithmic Expressions Evaluate the expression. A. Log 3 81 B. Log 5 0.04 C. Log ½ 8 D. Log 9 3

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Evaluate the expression. A. Log 4 64 B. Log 2 0.125 C. Log ¼ 256 D. Log 32 2 You Try

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Types of Logarithms A logarithm with base 10 is called the common logarithm (your calculator assumes base 10). log 10 x = log x (normally written without the base) The logarithm with base e is called the natural logarithm. We usually write this is “ ln ”. log e x = ln x

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Time for Calculators Evaluate. 1. log 5 2. ln 0.1

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Using Inverse Properties The expressions: g(x) = log b xf(x) = b x are inverses of one another. We can use this to simplify expressions. This means (get ready for this) g(f(x)) = log b b x = xand f(g(x)) = b log b x = x

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Examples Simplify the expression. 1. 10 log 10 2 2. log 3 9 x

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You Try Simplify the expression. 1. 10 log 10 5x 2. log 10,000 x

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Finding Inverses of Logarithmic Functions Find the inverse of the functions: 1. y = log 3 x 2. y = ln(x + 1)

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Graphing Logarithmic Functions The graph of y = log b (x – h) + k has the following characteristics: 1. The line x = h is a vertical asymptote 2. The domain is x>h, and the range is all real numbers. 3. If b > 1, the graph moves up to the right. If 0<b<1, the graph moves down and to the right.

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Example Graph

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Another Example Objective: 8.4 Logarithmic Functions17 Graph the function. State the domain and range. y = log 5 (x + 2)

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Homework page 490, 16-34 even 36, 37, 48-54 even, 56, 61, 65, 66, 69

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