Convert between logarithmic and exponential notation.

Evaluate logarithmic functions.

Graph logarithmic functions.

Use properties of logarithms to simplify expressions.

Logarithmic Functions

The logarithmic functions are the inverses of the exponential functions.

To more specific...

Let be a fixed positive real number not equal to 1. The logarithmic function with base-, denoted , is the inverse of the base- exponential function. That is,

Examples

because .

because .

Can you find two consecutive positive integers that bound ?

Your calculator should compute base-10 logarithms, often called common logs. Use your calculator to compute .

Properties of the Logarithmic Functions

Because the logs and exponentials are inverses, we must have:

for any real number

for any positive real number

Examples

In general, the logarithmic functions have the following properties.

Continuous and increasing

One-to-one ( Graph passes the horizontal line test.)

Domain: , i.e., all positive real numbers

Range: , i.e., all real numbers

is a vertical asymptote of the graph.

is the only -intercept of the graph.

is a point on the graph.

as , but it does so slowly.

Continuous and decreasing

One-to-one ( Graph passes the horizontal line test.)

Domain: , i.e., all positive real numbers

Range: , i.e., all real numbers

is a vertical asymptote of the graph.

is the only -intercept of the graph.

is a point on the graph.

as , but it does so slowly.

Examples

Discuss the graph of .

Discuss the graph of .

Discuss the graph of .

The Natural Logarithm

The base- logarithm is called the natural logarithm:

Your scientific calculator has built-in functions to compute base-10 and base- exponentials and logarithms.

Using the Properties of Logs

The properties of logarithms can be very useful when evaluating expressions and solving equations.