Exponential and Logarithmic Functions

Exponential functions model quantities that grow or decay at a rate proportional to their current value. The general form f(x) = a * b^x describes exponential growth when b > 1 and exponential decay when 0 < b < 1. The natural exponential function f(x) = e^x, where e is approximately 2.71828, is fundamental in calculus, finance, and the natural sciences.

Logarithmic functions are the inverses of exponential functions. The logarithm log_b(x) answers the question: to what power must the base b be raised to produce x? Common bases include 10 (common log), e (natural log, ln), and 2 (used in computer science). Properties of logarithms -- the product rule, quotient rule, power rule, and change of base formula -- are essential tools for simplifying expressions and solving exponential equations.

These functions appear throughout real-world applications: compound interest and continuous compounding in finance, population growth and radioactive decay in science, pH calculations in chemistry, the Richter scale for earthquakes, and decibel scales for sound intensity. Solving exponential and logarithmic equations requires fluency with logarithm properties and the ability to convert between exponential and logarithmic forms. This topic is central to AP Precalculus and serves as a gateway to calculus concepts like the derivative of e^x.

Practice a little. See where you stand.