Logarithm Calculator
Log to any base, natural log, antilog and solving for the base, with rules.
About this tool
Choose what to solve for: the logarithm y = log_b(x), the antilog x = b^y, or the base b. Pick base 10, e, 2 or a custom base and enter the numbers. The calculator shows the answer, the change-of-base working, and the same number's natural, common and binary logs. A table lists the main log rules.
What logarithms are and how to use them
A logarithm answers the question: what power do I raise the base to, to get this number? logโโ 1000 = 3 because 10ยณ = 1000, and logโ 8 = 3 because 2ยณ = 8. The common log (base 10) measures orders of magnitude and underlies decibels, pH and the Richter scale. The natural log ln uses base e โ 2.71828 and appears wherever growth or decay is continuous, from compound interest to radioactive half-lives. The binary log counts doublings, which is why it appears in computing. Any base can be computed with the change-of-base rule, log_b x = ln x รท ln b, which is how most calculators work. The antilog reverses a log, and you can also solve for an unknown base: if bโด = 81, then b = 3. Logs turn multiplication into addition, log(xy) = log x + log y, which is why slide rules worked. Logs are only defined for positive numbers, and the base must be positive and not equal to 1. The scientific calculator, the scientific notation converter and the half-life calculator use logs in practice. Compound interest, explained with real numbers shows logs and growth at work.
Frequently asked questions
What is log base 10 of 1000?
3, because 10 ร 10 ร 10 = 1000.
What is the difference between log and ln?
log usually means base 10 and ln means the natural log, base e โ 2.71828. Some programming languages use log for the natural log.
How do I calculate a log with a different base?
Use change of base: log_b x = ln x รท ln b. For example, logโ 81 = ln 81 รท ln 3 = 4.
Is my data saved?
No. Everything runs in your browser and nothing is stored or sent.