Logarithm Change of Base Formula
Learn the change of base formula for logarithms.
Convert any logarithm to base 10 or natural log for easy calculation, with worked examples.
The Formula
The change of base formula lets you evaluate a logarithm in any base using a calculator that only has log (base 10) or ln (natural log, base e) buttons. It works because all logarithm bases are proportionally related — you simply divide by the log of the base you want to convert from.
The most common forms used in practice are:
Variables
| Symbol | Meaning |
|---|---|
| logb(x) | Logarithm of x in the original base b (what you want to find) |
| x | The argument — the number you are taking the log of (must be > 0) |
| b | The original base (must be > 0 and ≠ 1) |
| c | The new base to convert to (any convenient base: 10 or e are most common) |
| log(x) | Common logarithm — base 10 (written without a subscript) |
| ln(x) | Natural logarithm — base e ≈ 2.71828 |
Example 1 — Basic Conversion
Evaluate log5(125) using the change of base formula.
Apply the formula: log5(125) = log(125) ÷ log(5)
log(125) = log(5³) = 3 × log(5) = 3 × 0.69897 = 2.09691
log(5) = 0.69897
log5(125) = 2.09691 ÷ 0.69897
log5(125) = 3 — confirming that 5³ = 125.
Example 2 — Non-Integer Result
Calculate log3(50).
log3(50) = log(50) ÷ log(3)
log(50) = 1.69897
log(3) = 0.47712
log3(50) = 1.69897 ÷ 0.47712 ≈ 3.561 — meaning 33.561 ≈ 50.
Example 3 — Using Natural Log
Calculate log2(1000) using the natural log (ln).
log2(1000) = ln(1000) ÷ ln(2)
ln(1000) = 6.90776
ln(2) = 0.69315
log2(1000) = 6.90776 ÷ 0.69315 ≈ 9.966 — meaning 29.966 ≈ 1000. (210 = 1024, confirming this is close to 10.)
Why the Formula Works
Let logb(x) = y. Then by the definition of a logarithm: by = x.
Take logc of both sides: logc(by) = logc(x)
Apply the power rule: y × logc(b) = logc(x)
Solve for y: y = logc(x) ÷ logc(b)
Substituting y = logb(x) gives the change of base formula.
Common Applications
- Computer science: Binary logarithms (log2) appear in algorithm complexity — big-O notation for binary search, tree height, and sorting algorithms. Calculators only have log10 and ln, so change of base is needed.
- Information theory: Shannon entropy uses log2 to measure information in bits. H = −Σ p(x) log2(p(x))
- Music: Pitch intervals are calculated in cents using log2. One semitone = 100 cents = log2(21/12) × 1200
- Finance: Continuous compound interest uses ln. The number of periods to double: t = ln(2) ÷ r (the Rule of 70 approximation)
- Chemistry: pH = −log10[H⁺]. Converting between Kₐ and pKₐ uses base-10 logs.