2Definition 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.
3Location of Base and Exponent in Exponential and Logarithmic Forms Logarithmic form: y = logb x Exponential Form: by = x.ExponentExponentBaseBase
4Text Example Write each equation in its equivalent exponential form. a. 2 = log5 x b. 3 = logb 64 c. log3 7 = ySolution 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.
5Text 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 5log3 9 = 2 because 32 = 9.3 to what power is 9?b. log3 9log2 16 = 4 because 24 = 16.2 to what power is 16?a. log2 16Logarithmic Expression EvaluatedQuestion Needed for EvaluationLogarithmic Expression
6Basic 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).
7Inverse 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.
8Text ExampleGraph 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.42811/21/4f (x) = 2x3-1-2x2431-1-2g(x) = log2 x81/21/4xReverse coordinates.
9Text Example cont.Graph f (x) = 2x and g(x) = log2 x in the same rectangular coordinate system.SolutionWe 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.-2-162345f (x) = 2xf (x) = log2 xy = x
10Characteristics 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.
11Properties of Common Logarithms General Properties Common Logarithms1. logb 1 = log 1 = 02. logb b = log 10 = 13. logb bx = x 3. log 10x = x4. b logb x = x log x = x
12Examples of Logarithmic Properties log b b = 1log b 1 = 0log 4 4 = 1log 8 1 = 03 log 3 6 = 6log = 32 log 2 7 = 7
13Properties of Natural Logarithms General Properties Natural Logarithms1. logb 1 = ln 1 = 02. logb b = ln e = 13. logb bx = x 3. ln ex = x4. b logb x = x 4. e ln x = x
14Examples of Natural Logarithmic Properties log e e = 1log e 1 = 0e log e 6 = 6log e e 3 = 3