1. Bases Other than e and Applications
Section 5.5 AP Calc

2. Definition of Exponential Function to Base a
If a is a positive real number (a ≠ 1) and x is any real number, then the exponential function to the base a is denoted by ax and is defined by If a = 1, then y = 1x = 1 is a constant function.

3. Example 1: The time in which a machine depreciates to one- half of its purchase price is 3 years. Find a model that yields the fraction of purchase price as a function of time and determine that fraction at time t0 = 6.

5. Write in log or exponential form:
Example 2:
log27 9 =
Write in log or exponential form:
23 = 8 b. log = -2
Solve: b.

6. Example 3: Solve the equation accurate to 3 decimal places. a. b.

8. Example 4: Find the derivatives of the following. a. b. c.

9. Example 5: find the integrals. a. b.
To integrate an exponential function with a base other than e, you can use the following formula or convert to base e and integrate.
Example 5: find the integrals.
a. b.

12. Example 6: In a group project in learning theory, a mathematical model for the proportion P of correct responses after n trials was found to be
Find the limiting proportion of correct responses as n approaches infinity.
Find the rate at which P is changing after n = 3 trials and n = 10 trials.