Inverse Functions and Logarithms Greg Kelly, Hanford High School, Richland, Washington Adapted by: Jon Bannon, Siena Colllege Photo by Vickie Kelly, 2004 Golden Gate Bridge San Francisco, CA

A relation is a function if: for each x there is one and only one y. A relation is a one-to-one if also: for each y there is one and only one x. In other words, a function is one-to-one on domain D if: whenever

To be one-to-one, a function must pass the horizontal line test as well as the vertical line test. one-to-onenot one-to-onenot a function (also not one-to-one)

Inverse functions: Given an x value, we can find a y value. Switch x and y : (eff inverse of x) Inverse functions are reflections about y = x. Solve for x :

Consider This is a one-to-one function, therefore it has an inverse. The inverse is called a logarithm function. Example: Two raised to what power is 16? The most commonly used bases for logs are 10: and e : is called the natural log function. is called the common log function.

is called the natural log function. is called the common log function. In calculus we will use natural logs exclusively. We have to use natural logs: Common logs will not work.

Properties of Logarithms Since logs and exponentiation are inverse functions, they “un-do” each other. Product rule: Quotient rule: Power rule: Change of base formula:

Example 6: $1000 is invested at 5.25 % interest compounded annually. How long will it take to reach $2500? We use logs when we have an unknown exponent years In real life you would have to wait 18 years. 