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.