Why Learn Logarithms?

A logarithm answers the question "what power do I raise the base to, to get this number?" That sounds abstract, but logarithms exist because some real-world quantities span an enormous range — from barely-detectable to overwhelming — and a logarithmic scale compresses that range into small, readable numbers.

The acidity of a solution is measured by its concentration of hydrogen ions, a number that can range from 1 to 0.0000000000001 mol/L. Instead of writing all those zeros, chemists use pH = −log[H⁺], which turns that huge range into a simple scale from 0 to 14. Each whole step on the pH scale represents a tenfold change in acidity — a solution with pH 3 is ten times more acidic than one with pH 4.

Earthquake magnitude is measured with M = log(A/A₀), where A is the amplitude of the ground motion. Because the scale is logarithmic, a magnitude 6 earthquake does not shake the ground twice as much as a magnitude 3 — it shakes it about 1,000 times more, since each whole step multiplies the ground motion by 10.

Sound intensity is measured in decibels using dB = 10 log(I/I₀). A quiet whisper and a jet engine differ in intensity by a factor of a trillion or more — far too unwieldy to compare directly. The decibel scale turns that into a manageable range, where every added 10 dB means the sound is 10 times more intense.

All three examples share the same idea: when a quantity can vary by many powers of ten, a logarithmic scale turns multiplication into addition, so huge ranges become small, easy-to-compare numbers. This is also why logarithms are the natural inverse of exponential growth and decay, and why they show up anywhere a quantity grows or shrinks by a constant factor rather than a constant amount.

Related lessons

• Frequency distribution and histograms