Change of Base Formula: Concepts, Formulas and Examples

The change-of-base formula states that log_a b = (log_c b)/(log_c a) for any convenient new base c. It lets you compute a logarithm in an awkward base using only the logarithms your calculator actually provides: base 10 or base e.

It matters because calculators offer just log (base 10) and ln (base e), yet science and computer science constantly need base 2, base 3, and other bases — this formula is the bridge between them.

Key Concepts

• Free choice of new base: c can be anything valid — in practice 10 or e, whichever your calculator has.
• Numerator and denominator roles: the original argument b goes on top, the original base a goes below.
• Reciprocal corollary: log_a b = 1/(log_b a), handy when the new base equals the old argument.
• Chain form: log_a b · log_b c = log_a c, which telescopes across intermediate bases.
• Value preserved: changing the base only rewrites the logarithm; its value never changes.

Formulas

The formula follows from writing b = a^(log_a b) and taking log_c of both sides. It is valid for a, b, c > 0 with a ≠ 1 and c ≠ 1.

• log_a b = (log_c b)/(log_c a)
• log_a b = 1/(log_b a)
• log_a b · log_b c = log_a c

Worked Example

Compute log_2 10 using common logarithms. Change to base 10: log_2 10 = log 10 / log 2 = 1/0.3010 ≈ 3.3219. Verify by exponentiation: 2^3.3219 = e^(3.3219 × ln 2) = e^(3.3219 × 0.6931) = e^2.3026 ≈ 10.000, which rounds to exactly 10. The computation checks out: raising 2 to 3.3219 returns the original argument 10.

Common Misconceptions

• log_a b = log b − log a. Subtraction is not division — the formula requires a quotient, and log b − log a actually equals log(b/a).
• Swapping numerator and denominator. (log_c a)/(log_c b) computes log_b a, the reciprocal of what you want.
• "Change of base changes the value." It only rewrites the expression; log_2 10 ≈ 3.3219 no matter which intermediate base you choose.

Real-World Applications

• Computing binary logarithms for computer science (bits, binary search depth).
• Converting between log and ln when differentiating or integrating in calculus.
• Evaluating logarithms on basic calculators that only have log and ln keys.
• Information theory, translating between bits (base 2) and nats (base e).

The change-of-base formula log_a b = (log_c b)/(log_c a) reduces every logarithm to ones your calculator knows. Keep the argument on top, the base below, and never confuse the quotient with a difference.