Logarithm

A logarithm is the inverse operation of exponentiation. The expression $\log_a b = c$ means exactly $a^c = b$ — the logarithm answers "to what power must I raise $a$ to get $b$?"

Common bases:

• $\log_{10}$ (common log) — used in pH, decibels, Richter scale.
• $\ln = \log_e$ (natural log) — calculus and continuous-growth models.
• $\log_2$ — computer science, information theory.

Key properties:

• $\log(xy) = \log x + \log y$ (turns product into sum)
• $\log(x^n) = n \log x$ (turns power into product)
• Change of base: $\log_a b = \frac{\log b}{\log a}$ for any reference base.

Logarithms compress huge ranges (Earth-Moon distance vs atom width) into tractable scales, and they linearise exponential data — that's why log-axis plots are so common in science.