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Published bySteven Nash Modified over 7 years ago

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Logarithmic Functions

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Definition of a Logarithmic Function For x > 0 and b > 0, b = 1, y = log b x is equivalent to b y = x. The function f (x) = log b x is the logarithmic function with base b.

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Location of Base and Exponent in Exponential and Logarithmic Forms Logarithmic form: y = log b x Exponential Form: b y = x. Exponent Base

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Text Example Write each equation in its equivalent exponential form. a. 2 = log 5 xb. 3 = log b 64c. log 3 7 = y SolutionWith the fact that y = log b x means b y = x, c. log 3 7 = y or y = log 3 7 means 3 y = 7. a. 2 = log 5 x means 5 2 = x. Logarithms are exponents. b. 3 = log b 64 means b 3 = 64. Logarithms are exponents.

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Evaluate a. log 2 16b. log 3 9 c. log 25 5 Solution log 25 5 = 1/2 because 25 1/2 = 5.25 to what power is 5?c. log 25 5 log 3 9 = 2 because 3 2 = 9.3 to what power is 9?b. log 3 9 log 2 16 = 4 because 2 4 = 16.2 to what power is 16?a. log 2 16 Logarithmic Expression Evaluated Question Needed for Evaluation Logarithmic Expression Text Example

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Basic Logarithmic Properties Involving One Log b b = 1 because 1 is the exponent to which b must be raised to obtain b. (b 1 = b). Log b 1 = 0 because 0 is the exponent to which b must be raised to obtain 1. (b 0 = 1).

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Inverse Properties of Logarithms For x > 0 and b 1, log b b x = xThe logarithm with base b of b raised to a power equals that power. b log b x = xb raised to the logarithm with base b of a number equals that number.

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Properties of Common Logarithms General PropertiesCommon Logarithms 1. log b 1 = 01. log 1 = 0 2. log b b = 12. log 10 = 1 3. log b b x = 03. log 10 x = x 4. b log b x = x 4. 10 log x = x

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Examples of Logarithmic Properties log 4 4 = 1 log 8 1 = 0 3 log 3 6 = 6 log 5 5 3 = 3 2 log 2 7 = 7

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Properties of Natural Logarithms General PropertiesNatural Logarithms 1. log b 1 = 01. ln 1 = 0 2. log b b = 12. ln e = 1 3. log b b x = 03. ln e x = x 4. b log b x = x 4. e ln x = x

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Examples of Natural Logarithmic Properties e log e 6 = e ln 6 = 6 log e e 3 = 3

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Problems Use the inverse properties to simplify:

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Characteristics of the Graphs of Logarithmic Functions of the Form f(x) = log b x The x-intercept is 1. There is no y-intercept. The y-axis is a vertical asymptote. (x = 0) If 0 1, the function is increasing. The graph is smooth and continuous. It has no sharp corners or edges. -2 6 2345 5 4 3 2 -2 6 f (x) = log b x b>1 -2 6 2345 5 4 3 2 -2 6 f (x) = log b x 0<b<1

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Graph f (x) = 2 x and g(x) = log 2 x in the same rectangular coordinate system. SolutionWe first set up a table of coordinates for f (x) = 2 x. Reversing these coordinates gives the coordinates for the inverse function, g(x) = log 2 x. 4 2 8211/21/4f (x) = 2 x 310-2x 2 4 310-2g(x) = log 2 x 8211/21/4x Reverse coordinates. Text Example

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Solution We now sketch the basic exponential graph. The graph of the inverse (logarithmic) can also be drawn by reflecting the graph of f (x) = 2 x over the line y = x. -2 6 2345 5 4 3 2 -2 6 f (x) = 2 x f (x) = log 2 x y = x Text Example Graph f (x) = 2 x and g(x) = log 2 x in the same rectangular coordinate system.

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Examples Graph using transformations.

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Domain of Logarithmic Functions Because the logarithmic function is the inverse of the exponential function, its domain and range are the reversed. The domain is { x | x > 0 } and the range will be all real numbers. For variations of the basic graph, say the domain will consist of all x for which x + c > 0. Find the domain of the following: 1. 2. 3.

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Sample Problems Find the domain of

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