Calculate the logarithm of a number in any base, including natural log.
logb(x) = ln(x) / ln(b). Natural log (ln) uses base e ≈ 2.71828.
A logarithm answers the question "what power do I need to raise this base to, to get this number?" — log₂(8) = 3 because 2³ = 8. That definition alone doesn't explain why logarithms show up constantly in real-world measurement, but the reason is genuinely practical: logarithmic scales compress enormous ranges of values into small, manageable numbers by turning multiplication into addition, which is exactly what's needed whenever a phenomenon naturally varies across many powers of ten.
The Richter scale for earthquakes is a direct example: a magnitude 6 earthquake isn't twice as strong as a magnitude 3 — it's roughly 1,000 times stronger, because each whole step up the scale multiplies the actual ground motion by a factor of 10. Decibels work the same way for sound intensity (dB = 10 log(I/I₀)) — a whisper and a jet engine differ in actual intensity by a factor of a trillion or more, a range far too unwieldy to compare directly, so the decibel scale compresses it down to a range where every added 10 dB represents a 10x increase in intensity. pH follows an identical logic for measuring acidity, based on hydrogen ion concentration. In every one of these cases, the underlying physical quantity spans many orders of magnitude, and a logarithmic scale is what makes that range usable and comparable on a simple number line rather than an unwieldy string of zeros.
A logarithm answers 'what power do I need to raise a given base to, to get this number?' log₂(8) = 3 because 2³ = 8. The logarithm and exponentiation are inverse operations of each other, similar to how subtraction reverses addition.
Because earthquake ground motion varies across an enormous range, and a logarithmic scale compresses that range into manageable whole numbers. Each whole step up the Richter scale represents roughly a 10x increase in actual ground motion — meaning a magnitude 6 earthquake is about 1,000 times stronger than a magnitude 3, not just twice as strong.
Decibels are calculated as dB = 10 log(I/I₀), comparing a sound's intensity to a reference level. Because actual sound intensity can differ by a factor of a trillion or more between a whisper and a jet engine, the logarithmic decibel scale compresses that into a practical range where every added 10 dB represents roughly a 10x increase in actual intensity.
They differ only in which base is used: log₁₀ (common log) uses base 10, ln (natural log) uses base e (≈2.71828), and log₂ uses base 2, common in computing. Each measures 'what power of the base gives this number,' just with a different base — and they can be converted between each other using the change-of-base formula.