NumberTally Log Calculator
Compute log base 10, natural log (ln), log base 2, or any custom base in seconds. Enter a number and a base below — all results update instantly as you type. Free, no sign-up.
Advanced options
The change-of-base formula
Every logarithm can be rewritten with natural logs: log_b(x) = ln(x) / ln(b). That is how calculators actually compute odd bases — and the steps box below shows the substitution with your numbers, plus a check by raising the base back to the result.
Comparing bases
The same number has a different logarithm in each base — smaller bases give bigger logs. The bar chart compares log₂, ln, and log₁₀ of your number side by side.
Frequently Asked Questions
What is the difference between log and ln?
"log" usually means base 10 (common log); "ln" is base e ≈ 2.71828 (natural log). They differ by a constant factor: ln(x) = log(x) × 2.302585.
How do you find the logarithm of a number with a custom base?
Use change of base: log_b(x) = ln(x)/ln(b). Pick "Custom…" above and enter any base.
Can you take the logarithm of a negative number or zero?
Not in real numbers — no power of a positive base gives zero or a negative. (Complex analysis extends it, with caveats.)
Why is the base of a logarithm never 1?
Because 1 raised to any power is 1, so "1 to what power equals x" has no answer (or infinitely many for x = 1).
How do logarithms relate to exponents?
They are inverses: log_b(x) = y means exactly b^y = x. The calculator verifies each answer this way.