Change of Base Formula

Pre-Calculus · Axiom Academy

Your calculator has only two log keys — here is the one identity that lets it evaluate a logarithm in any base. 1. One Logarithm, Any Base You Like The change of base formula says you may rewrite over any new base c , as long as and . The single value becomes a ratio of two logarithms that share that new base. Pick any base c — the result is the same Watch get evaluated two completely different ways: once with natural logs, , and once with common logs, . The numerator and denominator are different numbers each time — but the ratio lands on the exact same value , . 2. Why You Need It: Two Buttons, Any Base A calculator can only press the keys it has. With base‑10 and natural , the formula turns the impossible keystroke " " into two presses you can make and one division. Compute — this becomes the numerator. Compute — this becomes the denominator. Swap both s for s and the quotient is unchanged. Is 2.010 reasonable? Since 7^2 = 49 is just under 50 , should be a hair over 2 — and it is. The formula and your number sense agree. 3. When the Ratio Comes Out Exact Choosing the new base cleverly can make the answer a tidy fraction — no decimals at all. To evaluate , switch to base 2 , since 4 and 8 are both powers of 2 . Now both logs are whole numbers. The numerator: , because 2^3 = 8 — three factors of 2 . The denominator: , because 2^2 = 4 — two factors of 2 . — correct, but you'd never see the . — same value, now obviously a clean fraction.

This is the written version of the interactive lesson above. See the full Pre-Calculus course.