Logarithmic Functions

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.

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

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.

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

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).

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.

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 log x = x

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

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

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

Problems Use the inverse properties to simplify:

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 f (x) = log b x b> f (x) = log b x 0<b<1

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

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 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.

Examples Graph using transformations.

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:

Sample Problems Find the domain of