Chapter 4

4.2 – Logarithmic Functions

Objectives

1. Understand logarithmic functions are the inverse functions of exponential functions.
2. Know the graph of  and how it relates to the graph of .
3. Use properties to express exponential equations in logarithmic form and logarithmic functions in exponential form.
4. Use curve sketching methods to sketch graphs related to .
5. Perform calculator evaluations of  for various bases a and positive values of x.

Examples/Definitions

Logarithm with base a:  The inverse function of the exponential function  is denoted by  and is called a logarithmic function.

Natural Logarithm:  The inverse function of the natural exponential function  is denoted by  and is called the natural logarithm.

The key properties of  for base a > 1 are:

1.  has the domain
2.  has the range
3.  for any base a
4.  for any base a
5. the graph of   has the vertical asymptote x = 0 (y-axis)

The same properties can be extended to the natural logarithm function:

1.  has the domain
2.  has the range
3.  for any base a
4.  for any base a
5. the graph of   has the vertical asymptote x = 0 (y-axis)

Inverse Properties

 for all positive real numbers x

 for all real numbers x

The inverse properties also hold for the natural logarithmic function:

 for all positive real numbers x

 for all real numbers x