Unit 8 Exponential and Logarithmic Functions Their Graphs

Unit 8 Exponential and Logarithmic Functions & Their Graphs

Must pass the f(x) horizontal line test. x =3 Is this function one to one? Yes Does it have an inverse? Yes The inverse is called a logarithm function.

Logarithmic Function of base “a” Definition: Logarithmic function of base “a” - For x > 0, and a 1, y = logax if and only if x = ay Read as “log base a of x” f(x) = logax is called the logarithmic function of base a.

The most important thing to remember about logarithms is…

a logarithm is an exponent.

Therefore, all logarithms can be written as exponential equations and all exponential equations can be written as logarithmic equations.

Transformations y=loga(x-h)+k h left/right k up/down *negative outside reflects over x *negative inside parentheses reflects over y

Asymptote: y=loga(x-h)+k number moved left or right, use h number in ( ) (x – h) = 0 (solve for x)

Domain and Range Domain: (asymptote, ∞) (-∞, asymptote) if reflected across y- axis Range: Always all real numbers (- ∞, ∞)

End Behavior

Review: How do you find the inverse of a function? Application of what you know… What is the inverse of f(x) = 3 x? Rewrite the exponential as a logarithm… y = 3 x x = 3 y y = log 3 x f-1(x) = log 3 x

Find the inverse of the following exponential functions… f(x) = 2 x f-1(x) = log 2 x f(x) = 2 x+1 f-1(x) = log 2 x - 1 f(x) = 3 x- 1 f-1(x) = log 3(x + 1)

How to graph logarithmic functions Use calculator Make sure it’s in mathprint In TI-84, y=, Math, scroll down to logbase, Enter y = log 3(x+2)

Graphs of Logarithmic Functions g(x) = log 3(x + 2) Use change of base rule y= log 3(x +2) x Domain? (3, ) -1. 5 Range? (- , ) -1 1 Asymptotes? x = -2 7 y -. 63 0 1 2

Graphs of Logarithmic Functions g(x) = log 2(x + 3) - 1 Use change of base rule y= log 2(x + 3) – 1 x -2 Range? (- , ) 0 1 Asymptotes? x = -3 5 Domain? (-3, ) y -1. 58 1 2