Sec 4.1 Exponential Functions Objectives: To define exponential functions. To understand how to graph exponential functions.

Exponential Functions The exponential function with base a is defined for all real numbers x by where a > 0 and a  1.

Ex 1. Let and evaluate the following. a) b) c) d)

Graphs of Exponential Functions Ex 2. Graph each function. a)b)

HW p ,11,13

Sec 4.2 Logarithmic Functions Objectives: To define logarithmic functions. To understand properties of log functions.

Definition of the Logarithmic Function Let a be a positive number with a  1. The logarithmic function with base a, denoted by log a, is defined by

Ex 1. Express the equation in exponential form. a) log = 3 b) log 2 32 = 5 c) log = –1 d) log 16 4 = ½

Ex 2. Express the equation in logarithmic form. a) b) c) d)

PropertyReason 1log a 1 = 0 2log a a = 1 3log a a x = x 4a log a x = x Properties of Logarithms

Ex 3. Evaluate the expression. a) b) c)

Ex 4. Use the definition of the logarithmic function to find x. a) b) c)

The Natural Exponential Function The natural exponential function is the exponential function with base e.

The Natural Logarithmic Function The logarithm with base e is called the natural logarithm. ln x = log e x

Properties of Natural Logarithms PropertyReason 1ln 1 = 0 We must raise e to the power 0 to get 1. 2ln e = 1 We must raise e to the power 1 to get e. 3ln e x = x We must raise e to the power x to get e x. 4e ln x = x We must raise e to the power ln x to get x.

Ex 5. Evaluate the expression. a) b) c) d)

HW p odd