Logarithms: The Maths Behind Earthquakes, Sound, and Stars

A logarithm asks: 'what power do I need to raise a base to in order to get this number?' If 10² = 100, then log₁₀(100) = 2. Logarithms are the inverse of exponentiation. The three most important bases are 10 (common logarithm, used in science), e ≈ 2.718 (natural logarithm, used in calculus and many natural models), and 2 (used in computing). The key laws — log(ab) = log(a) + log(b), log(aⁿ) = n log(a) — turn multiplication into addition and powers into multiplication, which was their original appeal.

John Napier and the navigators

Scottish mathematician John Napier published his invention of logarithms in 1614. At the time, astronomers and navigators regularly needed to multiply 7- or 8-digit numbers together — a process that took many minutes of careful arithmetic and was prone to error. Logarithms converted these multiplications into additions, dramatically reducing the labour. Henry Briggs visited Napier and together they developed the base-10 system, publishing logarithm tables that were used for the next 350 years. The slide rule — a physical device that exploited logarithms to perform multiplication mechanically — was the engineer's calculator until the 1970s. Logarithms genuinely changed what was computationally feasible.

Logarithmic scales everywhere

The Richter scale (and its successor, moment magnitude) measures earthquake strength logarithmically: a magnitude 7 earthquake releases about 32 times the energy of a magnitude 6. The decibel scale for sound is logarithmic — 20 dB is not twice as loud as 10 dB, it's roughly 10 times as loud. The pH scale for acidity is logarithmic — each unit change represents a 10-fold change in hydrogen ion concentration. The sensitivity of human hearing and vision is roughly logarithmic — we perceive ratios of intensity rather than absolute differences. Star brightness is measured on a logarithmic magnitude scale. Shannon's formula for information entropy uses the natural logarithm. When something spans many orders of magnitude, a logarithmic scale is usually the right way to represent it.