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4.3 Rules of Logarithms

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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 = 1because 1 is the exponent to which b must be raised to obtain b. (b 1 = b). Log b 1 = 0because 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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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 /21/4f (x) = 2 x 310-2x g(x) = log 2 x 8211/21/4x Reverse coordinates. Text Example

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Graph f (x) = 2 x and g(x) = log 2 x in the same rectangular coordinate system. Solution We now plot the ordered pairs in both tables, connecting them with smooth curves. The graph of the inverse can also be drawn by reflecting the graph of f (x) = 2 x over the line y = x f (x) = 2 x f (x) = log 2 x y = x Text Example cont.

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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. If b > 1, the function is increasing. If 0 < b < 1, the function is decreasing. The graph is smooth and continuous. It has no sharp corners or edges.

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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 = x3. log 10 x = x 4. b log b x = x log x = x

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Examples of Logarithmic Properties log b b = 1 log b 1 = 0 log 4 4 = 1 log 8 1 = 0 3 log 3 6 = 6 log = 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 = x3. ln e x = x 4. b log b x = x 4. e ln x = x

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Examples of Natural Logarithmic Properties log e e = 1 log e 1 = 0 e log e 6 = 6 log e e 3 = 3

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4.3 Rules of Logarithms

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