Chapter 4: Exponential and Logarithmic Functions
Chapter 4 covers exponential and logarithmic functions — the math behind growth, decay, earthquakes, sound levels, and compound interest. These two types of functions are inverses of each other, and mastering their relationship unlocks a powerful set of problem-solving tools.
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Sections
4.1Exponential Functions
An exponential function is any function of the form f(x) = aˣ where a > 0 and a ≠ 1. The variable is in the exponent — that's what makes it different from polynomial functions. Exponential functions model anything that grows or decays at a constant percentage rate.
4.2The Natural Exponential Function
The number e ≈ 2.71828 is the base that makes calculus simplest. The function f(x) = eˣ appears everywhere in science and math. This section introduces e, explains why it's special, and covers continuous compounding.
4.3Logarithmic Functions
A logarithm answers the question: 'what power do I raise the base to in order to get this number?' It's the inverse of the exponential function. If you understand that log and exponential undo each other, you understand the most important thing about logs.
4.4Laws of Logarithms
Three simple laws let you expand, combine, and simplify logarithms. These are used constantly in solving equations and simplifying expressions in calculus and beyond.
4.5Exponential and Logarithmic Equations
This section is where the log laws pay off. To solve exponential equations, take logs of both sides. To solve logarithmic equations, convert to exponential form or exponentiate both sides. Always check for extraneous solutions.
4.6Modeling with Exponential Functions
All those equation-solving skills now get applied to real problems: population growth, radioactive decay, carbon dating, Newton's law of cooling, and more. The key is recognizing which model to use and solving for the unknown.
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