9.1, 9.3 Exponents and Logarithms

Graphs of Exponential Functions The graph of f(x) = bx has a characteristic shape. If b > 1, the graph goes uphill If 0 < b < 1, the graph goes downhill Domain is (–∞, ∞). Range is (0, ∞) Unless translated the graph has a y-intercept of (0,1) 24

Definition of a Logarithm A logarithm, or log, is defined in terms of an exponent: If 52=25 then log525=2 You can say that the log is the exponent we put on 5 to get 25 If bx=a, then logba=x

Logarithmic Functions x = 2y is an exponential equation. If we solved for “y” we would get a logarithmic equation. Here are the parts of each type of equation: Exponential Equation x = 2y Logarithmic Equation y = log2 x exponent /logarithm base number

Example: Solve loga64 = 2 a2 = 64 Example : Solve log5 x = 3 Rewrite in exponential form! loga64 = 2 base number exponent a2 = 64 a = + 8 → a = 8 Example : Solve log5 x = 3 Rewrite in exponential form: 53 = x x = 125

How do you graph a logarithmic function? Re-write it as an exponential function and make a T-chart: Example: Graph y = log3 x Rewrite as: x = 3y y = 3x x y 1/9 1/3 1 3 9 -2 -1 1 2 y = log3 x

Graphs of Logarithmic Functions The graph of f(x)=logbx has a characteristic shape. The domain of the function is {x | x >0} Unless translated, the graph has an x-intercept of 1. Note the domain and range! -1 1 2 3 4 5 6

The logarithm with base 10 is called the common logarithm (this is the one your calculator evaluates with the LOG button) The logarithm with base e is called the natural logarithm (this is the one your calculator evaluates with the LN button)

Examples. Evaluate each: a. log8 84 b. 6[log6 (3y – 1)] logb bx = x log8 84 = 4 blogb x = x 6[log6 (3y – 1)] = 3y – 1 Here are some IMPORTANT logarithm properties: 1. loga 1 = 0 because a0 = 1 2. loga a = 1 because a1 = a 3. loga ax = x because ax = ax