Common and Natural Logs

Algebra 2 · Axiom Academy

LESSON Common and Natural Logarithms Two special bases — (base 10) and (base e ) — and why your calculator gives each its own button. 1. The Common Log Reads a Power of 10 Write with no base showing and it means base 10: . It answers a single question — 10 to what power gives x ? The natural log is the logarithm base e , where . It answers — e to what power gives x ? — which is exactly the inverse of raising e to a power. The number e arises naturally in continuous growth. It's the base that makes calculus clean — the function e^ x is its own derivative. These bases aren't arbitrary — each one falls out of a whole family of real problems. Richter scale — earthquake magnitude Scientific notation — orders of magnitude Why base 10? These quantities span many powers of ten, and each +1 on the scale means in reality. Radioactive decay and half-life Calculus — derivatives and integrals Why base e ? Anything that grows or decays continuously follows e^ x , and is its natural inverse. Because they differ only in their base, one logarithm is a fixed constant multiple of the other. The change-of-base formula makes the link exact: The ratio is the same for every positive number: Find using the natural logarithm: You've met the two logarithms every calculator singles out — and seen exactly where each one belongs. Scroll up to revisit any step.

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