A logarithm answers the question: "What power must I raise the base to, to get this number?" If b^x = y, then log_b(y) = x.
Common Bases
- log₁₀ (common log): Used in pH, decibels, Richter scale
- ln (natural log, base e ≈ 2.718): Used in calculus, finance, growth models
- log₂ (binary log): Used in computer science, information theory
Key Properties
log(a×b) = log(a) + log(b) | log(a/b) = log(a) − log(b) | log(a^n) = n·log(a)
Why Logarithms Matter Beyond the Classroom
Logarithmic scales compress huge ranges into manageable numbers: the Richter scale (earthquakes), decibels (sound), and pH (acidity) are all logarithmic — each whole-number increase represents a 10× change in the underlying quantity. A magnitude 7 earthquake releases roughly 10× the energy of a magnitude 6, not just "one unit more."
Change of Base Formula
Most calculators only have log₁₀ and ln buttons, but you can find any base using: log_b(x) = ln(x) / ln(b). This is how log₂(8) (which equals 3) gets computed on a standard calculator that lacks a dedicated base-2 log function.