Mathematics July 22, 2026 · 9 min read

Change of Base Formula: Converting Between Logarithm Bases

Master the change of base formula log_b(x) = log_k(x)/log_k(b). Convert between any logarithm bases with worked examples.

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The change of base formula is one of the most practical logarithm properties. It allows you to compute any logarithm using only common log (base 10) or natural log (base e), which are the only buttons on most calculators.

The Formula

log_b(x) = log_k(x) / log_k(b)

Proof

Let y = log_b(x). Then b^y = x. Taking log_k of both sides: y × log_k(b) = log_k(x), so y = log_k(x)/log_k(b).

Worked Example

Convert log₂(8) to natural log:

  1. log₂(8) = ln(8)/ln(2)
  2. = 2.07944/0.69315 = 3
  3. Verify: 2³ = 8 —