Logarithm Calculator
Calculate logarithms in any base, including natural log and log base 10.
The Formula
Common log: log₁₀(x)
Natural log: ln(x) = log_e(x)
Change of base: log_b(x) = ln(x) / ln(b)
= log₁₀(10³)
= 3
ln(e⁵) = 5
log₂(32) = log(32)/log(2) = 5
What Is a Logarithm?
A logarithm answers the question: "To what power must we raise the base to get this number?" log_b(x) = y means bʸ = x. Logarithms are the inverse of exponential functions.
Common Logarithms
Common log (log or log₁₀): Used in pH calculations, the Richter scale, and decibels. Natural log (ln or logₑ): Uses Euler's number e ≈ 2.718; fundamental in calculus, compound interest, and population growth. Binary log (log₂): Used in computer science and information theory.
Frequently Asked Questions
What is a natural logarithm?
The natural logarithm (ln) is a logarithm with base e (Euler's number u2248 2.71828). It appears naturally in compound interest, population growth, radioactive decay, and many physics equations.
How do logarithms relate to exponents?
They are inverses: if 10u00b3 = 1000, then logu2081u2080(1000) = 3. If eu00b2 u2248 7.389, then ln(7.389) u2248 2.