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10.2 Logarithms and Logarithmic Functions Objectives: 1.Evaluate logarithmic expressions. 2.Solve logarithmic equations and inequalities.
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Logarithms The inverse of is In y is called the logarithm of x, usually written read “y equals log base b of x. Logs are a shortcut to solving for x or y. Let b and x be positive numbers, b≠1. The logarithm of x with base b is denoted and it defined as the exponent y that makes the equation true.
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Examples Write each equation in exponential form. 1. 2. Write each equation in logarithmic form. 1. 2.
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Logarithmic Functions This function is the inverse of the exponential function It has the following characteristics: 1.The function is continuous and one-to-one. 2.The domain is the set of all positive real numbers. 3.The y-axis is an asmyptote of the graph. 4.The range is the set of all real numbers 5.The graph contains the point (1,0). (The x- intercept is 1)
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Properties The following is true for all logarithms: Ex:
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Evaluating Log Expressions Write in exponential form. Rewrite with like- bases, set exponents equal to each other. Example: Evaluate:
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Logarithmic to Exponential Inequality If b>1, x>0, andthen x> If b>1, x>0, andthen 0<x< If x> If
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Property of equality for Log Functions If b is a positive number other than 1, then if and only if x=y. Example: if x=10
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Property of Inequality for Log Functions If b>1 then if and only if x>y and if and only if x<y. If then x<8 If then x>3
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Examples Solve. 1. 2. 3.
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Homework p. 536 22-40 even 48-60 even
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