Logarithm Calculator
Calculate logarithms
Frequently Asked Questions
What is a logarithm?
A logarithm answers the question: what exponent gives a certain result? log_b(x) = y means b^y = x. For example, log₂(8) = 3 because 2³ = 8. Common bases are 10 (log), 2 (lg), and e (ln).
What is the difference between log, ln, and log₂?
log (or log₁₀) uses base 10 — common in science and engineering. ln (natural log) uses base e ≈ 2.718 — used in calculus and continuous growth. log₂ uses base 2 — used in computer science and information theory.
What are the key logarithm rules?
Product: log(ab) = log(a) + log(b). Quotient: log(a/b) = log(a) - log(b). Power: log(a^n) = n × log(a). Change of base: log_b(x) = log(x) / log(b). These rules simplify complex calculations.
How do I change the base of a logarithm?
Use the change of base formula: log_b(x) = ln(x) / ln(b) or equivalently log(x) / log(b). For example, log₃(81) = ln(81) / ln(3) = 4.394 / 1.099 = 4.
Where are logarithms used in real life?
Decibel scale (sound intensity), Richter scale (earthquakes), pH scale (acidity), compound interest calculations, population growth models, and algorithm complexity analysis (O(log n)) all use logarithms.