Logarithm Calculator

Calculate log₁₀, natural log (ln), log₂, and logarithms for any base.

Logarithm Calculator

log₁₀(1000) =

3

log₁₀(1000)3
ln(1000)6.907755
log₂(1000)9.965784

Logarithm Calculator Formula & How It Works

logₐ(x) = ln(x) / ln(a) | ln(x) = logₑ(x) | log(x) = log₁₀(x)
  • log₁₀ (common log): base 10, used in pH, decibels, Richter scale
  • ln (natural log): base e ≈ 2.71828, used in calculus and finance
  • log₂ (binary log): base 2, used in computer science
  • Change of base: logₐ(x) = log(x)/log(a)

Logarithms are the inverse of exponentials: if aˣ = y, then logₐ(y) = x. The natural log (ln) uses base e and appears naturally in growth/decay equations, compound interest, and calculus. The common log (log₁₀) is used in scientific measurements. Binary log (log₂) counts bits required to represent n values.

Logarithm Calculator FAQs

What is the difference between log and ln?

log (common logarithm) uses base 10: log(100) = 2 because 10² = 100. ln (natural logarithm) uses base e ≈ 2.718: ln(e²) = 2. In mathematics, 'log' often means ln; in engineering it typically means log₁₀. Always check context.

How are logarithms used in real life?

Richter scale (earthquake magnitude), decibels (sound intensity), pH scale (acidity), compound interest calculations, information theory, computer algorithm complexity (O(log n)), signal processing, and population growth models.

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