How to use the Logarithm Calculator
- Enter the number whose logarithm you need, it must be positive.
- Enter the base, or pick 10, e or 2.
- Read the result and the equivalent exponential statement.
- Use the change-of-base display to reproduce the result on any calculator.
How the calculation works
log_b(x) = y means b^y = x. Any base can be computed from any other by change of base: log_b(x) = ln(x) ÷ ln(b), which is how calculators offering only log₁₀ and ln handle base 2 or base 7. The domain is strictly positive: logs of zero and negative numbers are undefined over the reals, since no real exponent of a positive base produces them.
The three product rules do the heavy lifting: log(xy) = log x + log y, log(x/y) = log x − log y, and log(xⁿ) = n log x. They turn multiplication into addition, which is why logs powered slide rules and still underpin log-scale charts. Base choice is conventional by field. 10 for decibels and pH, e for growth and decay, 2 for information and algorithm complexity.
log_b(x) = ln(x) / ln(b) ; b^(log_b x) = xSource: Standard logarithm identities and the change-of-base theorem.
Worked example
How many doublings take a population from 400 to 51,200?
- Growth factor = 51,200 / 400 = 128.
- Doublings = log₂(128).
- Change of base: ln(128) / ln(2) = 4.8520 / 0.6931.
- = 7.
Exactly 7 doublings, and 2⁷ = 128 confirms it.
Frequently asked questions
What is the difference between log and ln?+
log usually means base 10; ln is base e ≈ 2.71828, the natural base for continuous growth.
Why can't I take the log of a negative number?+
No real exponent of a positive base yields a negative result. Complex logarithms extend the definition, but not over the reals.
What is log₂ used for?+
Information theory and algorithm analysis, bits, entropy and binary search depth.
What does log(1) equal?+
Zero in every base, because any base raised to the power 0 is 1.
Last reviewed August 31, 2026. We review this page whenever the underlying formula, tax year, published rate or standard changes.