Log Calculator
Free logarithm calculator — log to any base, plus natural log (ln), log₁₀, and log₂, with verification and change-of-base.
Everyday Uses
Algebra and precalc
Any base, change-of-base shown, answer verified by raising the base back — homework with receipts.
pH and decibels
Chemistry pH and audio dB are logarithms — compute the underlying math directly.
Computer science
log₂ answers "how many halvings?" — binary search steps, tree depths, and bits needed.
Growth timelines
How many periods until an investment doubles? Logs solve for the exponent.
Frequently Asked Questions
What is a logarithm?
The logarithm answers: "what exponent turns the base into this number?" log₁₀(1000) = 3 because 10³ = 1000. Logarithms are the inverse of exponentiation, turning multiplication into addition — the property that made slide rules work and still powers decibels, pH, and earthquake magnitude scales.
What is the difference between ln, log₁₀, and log₂?
They differ only in base: ln uses e ≈ 2.71828 (natural sciences, calculus, continuous growth), log₁₀ uses 10 (engineering, pH, decibels), and log₂ uses 2 (computer science, information theory — "how many times can you halve this?"). Any log can be converted to another base by dividing: log_b(x) = ln(x)/ln(b).
Why can't I take the log of a negative number or zero?
Because no real exponent turns a positive base into a negative number or zero — 10 to any power is always positive. log(0) approaches negative infinity, and log of negatives requires complex numbers (beyond this calculator's scope). The domain of every real logarithm is strictly positive numbers.
What is the change-of-base formula used for?
log_b(x) = ln(x) / ln(b) lets you compute a logarithm in any base using only the ln or log₁₀ button on a basic calculator. Need log₇(50)? Compute ln(50)/ln(7) ≈ 2.011. This calculator applies it automatically for whatever base you enter.