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Bases Other than e and Applications

Explore the definition and applications of exponential functions with bases other than e in this comprehensive guide. Learn how to find derivatives, integrals, and solve equations involving exponential functions effectively. Discover practical examples and calculations to deepen your understanding.

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Bases Other than e and Applications

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  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.

  4. Example 2: • log27 9 = • Write in log or exponential form: • 23 = 8 b. log10 0.01 = -2 • Solve: • b.

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

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

  7. 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.

  8. 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.

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