4.2 Logarithmic Functions Think back to “inverse functions”. How would you find the inverse of an exponential function such as: Or more generally:

Definition of a Logarithmic Function For x > 0 and b > 0, b = 1, y = logb x is equivalent to by = x. (Notice that this is the INVERSE of the exponential function f(x) = y = bx) The function f (x) = logb x is the logarithmic function with base b.

Location of Base and Exponent in Exponential and Logarithmic Forms Logarithmic form: y = logb x Exponential Form: by = x. Exponent Exponent Base Base To convert from log to exponential form, start with the base, b, and move clockwise across the = sign: b to the y = x.

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, 2 = log5 x means: . b. 3 = logb 64 means: c. log3 7 = y or y = log3 7 means: .

log25 5 = ____ because 25___ = 5. 25 to what power is 5? c. log25 5 Evaluate a. log2 16 b. log3 9 c. log25 5 Solution log25 5 = ____ because 25___ = 5. 25 to what power is 5? c. log25 5 log3 9 = ____ because 3__ = 9. 3 to what power is 9? b. log3 9 log2 16 = ____ because 2__ = 16. 2 to what power is 16? a. log2 16 Logarithmic Expression Evaluated Question Needed for Evaluation Logarithmic Expression

Basic Logarithmic Properties Involving One logb b = because ____is the exponent to which b must be raised to obtain b. (b__ = b). logb 1 = because ____ is the exponent to which b must be raised to obtain 1. (b__ = 1).

Inverse Properties of Logarithms For b>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. That is: since logarithmic and exponential functions are inverse functions, if they have the SAME BASE they “cancel each other out”.

Properties of Logarithms General Properties Common Logarithms* 1. logb 1 = 0 1. log 1 = 0 2. logb b = 1 2. log 10 = 1 3. logb bx = x 3. log 10x = x 4. b logb x = x 4. 10 log x = x * If no base is written for a log, base 10 is assumed. If it says ln, that means the “natural log” and the base is understood to be e. Natural Logarithms* 1. ln 1 = 0 2. ln e = 1 3. ln ex = x 4. e ln x = x

ln e = ln 1 = e ln 6 = ln e 3 = log 4 4 = log 8 1 = 3 log 3 6 = Ex: log 4 4 = log 8 1 = 3 log 3 6 = log 5 5 3 = 2 log 2 7 = ln e = ln 1 = e ln 6 = ln e 3 =

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.

Continued… 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 Where is the asymptote for the exponential function, where is it now for the log function? What happened to the exponential functions y-intercept?

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.

Log Graphs using Transformations shift c up c>0 shift c down c<0 shift c right c>0 shift c left c<0 reflect about y-axis reflect about x-axis

Also do p 421 # 18, 64, 78, 82, 102, if time 119 18. Write in equivalent log form:

64. Graph f(x) = logx, then use transformations to graph g(x)= 2-logx 64. Graph f(x) = logx, then use transformations to graph g(x)= 2-logx. Find the asymptote(s), domain, range, and x- and y- intercepts.

78. Find the domain of f(x) = log (7-x) 82. Evaluate without a calculator: log 1000.

102. Write in exponential form and solve: