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.