Logarithms
A Comprehensive Lesson for Pre-University Algebra
1. Introduction & Motivation
Before calculators existed, multiplying large numbers was painstaking work. In 1614, the Scottish mathematician John Napier published a revolutionary idea: the logarithm — a tool that reduces multiplication and division to simple addition and subtraction.
Core Idea: Logarithms compress multiplication into addition and exponential growth into linear steps. Conversely, exponentials expand addition into multiplication.
Today, logarithms are indispensable across science, engineering, finance, and computer science. This lesson takes you from the basic definition through key properties, worked examples, real-world applications, and common pitfalls.
2. Prerequisites: Exponential Functions (Quick Review)
Before defining logarithms, recall the exponential function. For a base $b > 0$ with $b \neq 1$:
$$f(x) = b^x$$
Key exponent rules you should already know:
| Rule | Formula | Example |
|---|---|---|
| Zero Exponent | $b^0 = 1$ | $5^0 = 1$ |
| Identity | $b^1 = b$ | $5^1 = 5$ |
| Product of Powers | $b^m \cdot b^n = b^{m+n}$ | $2^3 \cdot 2^4 = 2^7 = 128$ |
| Quotient of Powers | $\dfrac{b^m}{b^n} = b^{m-n}$ | $\dfrac{3^5}{3^2} = 3^3 = 27$ |
| Power of a Power | $(b^m)^n = b^{mn}$ | $(2^3)^2 = 2^6 = 64$ |
Key Insight: If the logarithm is understood as the inverse of the exponential function, then the properties of logarithms will be seen as naturally flowing out of these rules for exponents.