Logarithm Calculator

Free online logarithm calculator. Compute log base 10 (log₁₀), natural log (ln), log base 2, and any custom base logarithm instantly.

Logarithms are the inverse of exponentiation. If bˣ = y, then log_b(y) = x. This calculator supports three modes: common logarithm (log₁₀), natural logarithm (ln, base e ≈ 2.718), and any custom base. Logarithms are essential in science, engineering, music, and finance.

Key Logarithm Properties

  • Product rule: log(a × b) = log(a) + log(b)
  • Quotient rule: log(a / b) = log(a) − log(b)
  • Power rule: log(aⁿ) = n × log(a)
  • Change of base: log_b(x) = ln(x) / ln(b)
  • log₁₀(1000) = 3 because 10³ = 1000

Practical Uses

The Richter scale for earthquakes and decibels for sound are both logarithmic scales — each step of 1 represents a 10× increase in intensity. In finance, logarithms are used to calculate continuously compounded interest. In computer science, log₂ measures the number of binary digits needed to represent a number.

Related Calculators