5.2 Logarithmic Functions & Their Graphs Goals— Recognize and evaluate logarithmic functions with base a Graph Logarithmic functions Recognize, evaluate, and graph natural logs Use logarithmic functions to model and solve real-life problems.

f(x) = 3 x Is this function one to one? Horizontal Line test? Does it have an inverse?

Logarithmic function with base “a” Def’n of Logarithmic function with base “a” For x > 0, a > 0, and a  1, y = log a x if and only if x = a y The function given by f(x) = log a x read as “log base a of x” is called the logarithmic function with base a.

Write the logarithmic equation in exponential form log 3 81 = 4 log 16 8 = 3/4 Write the exponential equation in logarithmic form 8 2 = = 1/ = /4 = 8 log 8 64 = 2 log 4 (1/64) = -3

Evaluating Logs f(x) = log 2 32 f(x) = log 4 2 f(x) = log 3 1 f(x) = log 10 (1/100) Step 1- rewrite it as an exponential equation. 2 y = 32 Step 2- make the bases the same. 2 y = 2 5 Therefore, y = 5 4 y = 2 2 2y = 2 2y = 2 y = 1 3 y = 1 y = 0 10 y = 1/ y = y = -2

Evaluating Logs on a Calculator f(x) = log x when x = 10 f(x) = 1 when x = 1/3 f(x) = when x = 2.5 f(x) =.3979 when x = -2 f(x) = ERROR!!! Why??? You can only use a calculator when the base is 10

Properties of Logarithms log a 1 = 0 because a 0 = 1 log a a = 1 because a 1 = a log a a x = x and a log a x = x log a x = log a y, then x = y

Simplify using the properties of logs log 4 1 log  7  7 6 log 6 20 Rewrite as an exponent 4 y = 1 So y = 0 Rewrite as an exponent  7 y =  7 So y = 1

Use the 1-1 property to solve log 3 x = log 3 12 log 3 (2x + 1) = log 3 x log 4 (x 2 - 6) = log 4 10 x = 12 2x + 1 = x x = -1 x = 10 x 2 = 16 x =  4

f(x) = 3 x Graphs of Logarithmic Functions So, the inverse would be g(x) = log 3 x Make a T chart Domain— Range? Asymptotes?

Graphs of Logarithmic Functions g(x) = log 4 (x – 3) Make a T chart Domain— Range? Asymptotes?

Graphs of Logarithmic Functions g(x) = log 5 (x – 1) + 4 Make a T chart Domain— Range? Asymptotes?

Natural Logarithmic Functions The function defined by f(x) = log e x = ln x, x > 0 is called the natural logarithmic function.

Evaluate f(x) = ln x when x = 2 f(x) =.6931 when x = -1 f(x) = Error!!! Why???

Properties of Natural Logarithms ln 1 = 0 because e 0 = 1 ln e = 1 because e 1 = e ln e x = x and e lnx = x (Think…they are inverses of each other.) If ln x = ln y, then x = y

Use properties of Natural Logs to simplify each expression ln (1/e) = ln e -1 = -1 e ln 5 = 5 2 ln e = 2

Graphs of Natural Logs g(x) = ln(x + 2) Make a T chart Domain— Range? Asymptotes? 2 Undefined 3 4

Graphs of Natural Logs g(x) = ln(2 - x) Make a T chart Domain— Range? Asymptotes? 2 Undefined 1 0