Change of Base Formula

Algebra 2 · Axiom Academy

Your calculator has only two log keys — and . Here's the one formula that turns them into a logarithm of any base. A logarithm asks a question. asks: 3 raised to what power gives 27? You could reason it out here — but your calculator has no base-3 key. It offers only (base 10) and (base e ). The formula isn't arbitrary — it falls straight out of the definition of a logarithm. Let be the value we're after. Since , the definition of a logarithm says a^y = x . Now take the logarithm — in any base c — of both sides and watch the exponent come down. That last line is exactly the change of base formula: Pick any base c — base 10 (the key) and base e (the key) are the handy choices. Use it to evaluate with the ordinary base-10 key. This is why a scientific calculator needs just two logarithm keys. Every base collapses to a single division you can key in. 5. Any Base Gives the Same Answer The intermediary base c is a free choice. Route through base 10, base e , or even base 2 — the ratio lands on the same value every time. You can now turn any logarithm into a ratio your calculator can evaluate — and you know exactly why the trick works.

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