Logarithm Basics
A logarithm answers: "to what power must the base be raised to get x?" If log₁₀(100) = 2, that means 10^2 = 100. The natural log (ln) uses base e ≈ 2.718, and appears throughout calculus and growth modeling.
log_b(x) = y means b^y = x
Change of base: log_b(x) = ln(x) / ln(b)
| Expression | Value | Meaning |
|---|---|---|
| log(1000) | 3 | 10^3 = 1000 |
| ln(e) | 1 | e^1 = e |
| log₂(32) | 5 | 2^5 = 32 |
Frequently Asked Questions
log (common logarithm) uses base 10 and is used in pH, decibels, and the Richter scale. ln (natural logarithm) uses base e and appears in calculus, exponential growth/decay, and finance formulas like continuous compounding.
pH = -log[H+] in chemistry; decibels use log10 for sound intensity; the Richter scale uses log10 for earthquake magnitude; compound interest uses ln for continuous compounding; information theory uses log2 (bits).