Logarithm Calculator
Calculate logarithms with any base using the change-of-base formula.
Returns ln (base e), log (base 10), and lb (base 2) with step-by-step working.
Logarithms are the inverse of exponentiation — if bⁿ = x, then log_b(x) = n. They answer the question: “To what power must the base be raised to produce this number?”
Core definitions: logb(x) = n means bⁿ = x
Common logarithm types:
- Common log (log₁₀ or log): base 10 — used in pH, decibels, Richter scale
- Natural log (ln): base e ≈ 2.71828 — used in calculus, growth/decay, finance
- Binary log (log₂): base 2 — used in computer science, information theory
Key logarithm rules:
| Rule | Formula | Example |
|---|---|---|
| Product | log(a×b) = log(a) + log(b) | log(100) = log(10) + log(10) = 2 |
| Quotient | log(a/b) = log(a) − log(b) | log(10/2) = 1 − log(2) |
| Power | log(aⁿ) = n × log(a) | log(1000) = 3 × log(10) = 3 |
| Change of base | log_b(x) = ln(x) ÷ ln(b) | log₂(8) = ln(8)/ln(2) = 3 |
Key values to memorize:
- log₁₀(10) = 1 | log₁₀(100) = 2 | log₁₀(1000) = 3
- ln(1) = 0 | ln(e) = 1 | ln(e²) = 2
- log₂(2) = 1 | log₂(4) = 2 | log₂(1024) = 10
Worked example — decibel calculation: Sound intensity formula: dB = 10 × log₁₀(I ÷ I₀) Where I₀ = 10⁻¹² W/m² (threshold of hearing) Normal conversation (I = 10⁻⁶ W/m²): dB = 10 × log₁₀(10⁻⁶ ÷ 10⁻¹²) = 10 × log₁₀(10⁶) = 10 × 6 = 60 dB
Natural log in finance (continuous compounding): A = P × e^(r×t) → t = ln(A/P) ÷ r Time to double at 7% continuously: t = ln(2) ÷ 0.07 = 0.693 ÷ 0.07 = 9.9 years