Logarithm Calculator
Calculate the logarithm of any positive number with any base using this free online tool. Choose from common logarithm (base 10), natural logarithm (base e), binary logarithm (base 2), or enter your own custom base.
A logarithm answers the question: 'To what power must the base be raised to produce a given number?' If b^y = x, then log_b(x) = y.
Common Types of Logarithms:
- Common Logarithm (log₁₀): Base 10. Used in science, engineering, and the decibel scale. For example, log₁₀(1000) = 3 because 10^3 = 1000.
- Natural Logarithm (ln): Base e (approximately 2.71828). Essential in calculus, physics, and continuous growth/decay models. For example, ln(e^2) = 2.
- Binary Logarithm (log₂): Base 2. Fundamental in computer science, information theory, and algorithm analysis. For example, log₂(256) = 8.
Key Logarithm Properties:
- log_b(m x n) = log_b(m) + log_b(n) (Product Rule)
- log_b(m / n) = log_b(m) - log_b(n) (Quotient Rule)
- log_b(m^k) = k x log_b(m) (Power Rule)
- log_b(1) = 0 for any valid base
- log_b(b) = 1 for any valid base
- Change of base: log_b(x) = log_c(x) / log_c(b)
Logarithms are the inverse of exponentiation and are indispensable in fields ranging from seismology (Richter scale) to music (frequency intervals) to finance (continuously compounded interest).