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
How we build and check this calculator
This calculator runs entirely in your browser, so the numbers you enter stay on your device. The math behind it is written by hand and tested against worked examples and standard references before the page goes live.
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