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