Common and Natural Logarithms

Pre-Calculus · Axiom Academy

LESSON Common and Natural Logarithms Two logarithms show up so often they earn their own names — "log" for base 10, "ln" for base e . A logarithm answers one question: what exponent turns the base into this number? The "common" log fixes that base at 10 , and the "natural" log fixes it at e . Same machine — two favorite settings. Common log — drop the subscript, base is 10 Natural log — "ln" means base e When you see with no subscript, it means base 10 — the log button on your calculator. Because each step up a power of 10 adds exactly 1 to the exponent, there's a quick trick: the log of a power of 10 is the exponent. The notation stands for logarithmus naturalis . It uses base , so is the exact inverse of e^ x : feed it e^ x and you get x back. Geometrically, undoes the natural-growth curve. because e^ 1 = e — the natural log of its own base is always 1 , exactly like for the common log. Each base wins in a different arena. Base 10 matches how we write numbers, so it tames quantities that span many powers of ten. Base e is the rate constant of continuous growth, so it's the language of calculus, interest, and decay. A log is just the mirror image of its exponential across the line y = x . Decibels (sound), the pH scale (acidity), and the Richter scale (earthquakes) all compress wide ranges onto a base-10 log scale. Continuous compound interest, radioactive decay and half-life, and the derivatives/integrals of calculus all run on base e .

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