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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 = logb x is equivalent to by = x. The function f (x) = logb 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 = logb x Exponential Form: by = x. Exponent Exponent Base Base

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**Text Example Write each equation in its equivalent exponential form.**

a. 2 = log5 x b. 3 = logb 64 c. log3 7 = y Solution With the fact that y = logb x means by = x, a. 2 = log5 x means 52 = x. Logarithms are exponents. b. 3 = logb 64 means b3 = 64. Logarithms are exponents. c. log3 7 = y or y = log3 7 means 3y = 7.

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**Text Example Evaluate a. log2 16 b. log3 9 c. log25 5 Solution**

log25 5 = 1/2 because 251/2 = 5. 25 to what power is 5? c. log25 5 log3 9 = 2 because 32 = 9. 3 to what power is 9? b. log3 9 log2 16 = 4 because 24 = 16. 2 to what power is 16? a. log2 16 Logarithmic Expression Evaluated Question Needed for Evaluation Logarithmic Expression

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**Basic Logarithmic Properties Involving One**

Logb b = 1 because 1 is the exponent to which b must be raised to obtain b. (b1 = b). Logb 1 = 0 because 0 is the exponent to which b must be raised to obtain 1. (b0 = 1).

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**Inverse Properties of Logarithms**

For x > 0 and b 1, logb bx = x The logarithm with base b of b raised to a power equals that power. b logb x = x b raised to the logarithm with base b of a number equals that number.

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Text Example Graph f (x) = 2x and g(x) = log2 x in the same rectangular coordinate system. Solution We first set up a table of coordinates for f (x) = 2x. Reversing these coordinates gives the coordinates for the inverse function, g(x) = log2 x. 4 2 8 1 1/2 1/4 f (x) = 2x 3 -1 -2 x 2 4 3 1 -1 -2 g(x) = log2 x 8 1/2 1/4 x Reverse coordinates.

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Text Example cont. Graph f (x) = 2x and g(x) = log2 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) = 2x over the line y = x. -2 -1 6 2 3 4 5 f (x) = 2x f (x) = log2 x y = x

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**Characteristics of the Graphs of Logarithmic Functions of the Form f(x) = logbx**

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 Properties Common Logarithms 1. logb 1 = log 1 = 0 2. logb b = log 10 = 1 3. logb bx = x 3. log 10x = x 4. b logb 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 Properties Natural Logarithms 1. logb 1 = ln 1 = 0 2. logb b = ln e = 1 3. logb bx = x 3. ln ex = x 4. b logb 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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Q Exponential functions f (x) = a x are one-to-one functions. Q (from section 3.7) This means they each have an inverse function. Q We denote the inverse.

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