Rules of logarithms
Interactive reference of logarithm rules (product, quotient, power, change of base) with examples.
Why the rules hold
The rules of logarithms follow from the rules of powers, because a logarithm is the inverse of exponentiation. For example, the rule logₐ(x·z) = logₐx + logₐz corresponds to the fact that when multiplying powers with the same base, the exponents add. It is exactly this property – a logarithm turns multiplication into addition – that historically made logarithms a powerful tool for fast calculation.
Change of base
Calculators usually only offer ln and log₁₀. A logarithm with any other base can therefore be computed via the change-of-base rule: logₐx = ln x / ln a (or with log₁₀ instead of ln). For example log₂ 8 = ln 8 / ln 2 = 3.
Tip: the rules hold for any valid base (positive, ≠ 1) and positive arguments.