Math Calculators
Logarithm Calculator
A logarithm is an exponent read backwards. log₁₀(1000) asks how many times 10 is multiplied by itself to reach 1000, and the answer is 3. Enter any two of the three quantities — the base, the argument and the logarithm — and this page solves the third, printing the answer exactly when it is exact.
Calculator
Formula
The whole page rests on one definition: log_b(x) = y means exactly b^y = x. Everything else follows from it. log_b(1) = 0 because the empty product is 1, and log_b(b) = 1 because one factor of b is b. The product rule is that definition read backwards — b^(m+n) = b^m · b^n, so the exponent that produces a product is the sum of the exponents that produce each factor — and the quotient, power and change-of-base rules come from the same reading. Two things are stated rather than left to the reader. A result is exact when raising the base to it reproduces the argument exactly, so log₁₀(1000) prints 3 and never 2.9999999999999996; a candidate is only even considered exact when it sits within 1e-12 of the value that would make it exact, and the exact check then confirms it. Anything else is rounded to ten significant digits and marked as rounded, so log₁₀(2) reads 0.3010299957.
About This Calculator
What It Does
This page answers one relation — log_b(x) = y — in whichever direction you asked for. In the default direction it evaluates the logarithm from a base and an argument, with the two named bases log₁₀ and ln beside the base control. In the second it recovers the base from an argument and a logarithm. In the third it reverses the logarithm and returns the argument, which is the quantity the competitor pages sell as an antilog. Under the result sits a fixed table of the six logarithm rules.
Worked Example
The page opens in Find the logarithm with the base set to log₁₀ and the argument at 1000, and answers 3 — exactly, because 10 × 10 × 10 = 1000. Switch Find to the base and the same numbers read the other way: with the argument at 1000 and the logarithm at 3, the page answers 10, since 10 cubed is 1000. Switch to the argument and it answers 1000 back, because 10³ = 1000.
Real-World Usage
Use it to read a quantity written as a power of ten — an earthquake magnitude, a sound level, an acidity, a pH — or to recover the number of periods a quantity has been growing for. It is the page to reach for when a growth or decay problem gives you the final value and the rate and asks for the time, and when a number is written as a logarithm and you need the plain value back.
Local Context
A logarithm is a property of the numbers and not of the country, so nothing here changes with your location and the page serves one set of answers everywhere — the arithmetic is consolidated to a single page per language. Two conventions are worth knowing. In the input fields the decimal separator is the point, as in 0.30103 and 2.718281828, whatever separator your language uses in ordinary prose; the result is formatted in your language's own convention, so the same answer reads 0,3010299957 in German and 0.3010299957 in English. And an exact answer is printed in full, while a rounded one is marked as rounded next to its ten-significant-digit value.
Sources and Methodology
The six logarithm rules on this page are the standard identities that follow from the definition, and each row's derivation is stated in the calculator module itself. The distinction between the decimal and the natural logarithm, and the convention that a logarithm without a written base means base 10, are the conventions the notation carries in every language this page serves. The exactness window and the ten-significant-figure display rule are this page's own and are stated on it, not quoted from a standard.
Last reviewed: October 2026.
Frequently Asked Questions
Why does log₁₀(1000) print 3 and not 2.9999999999999996?
Because the two are not the same answer. 1000 is exactly 10 × 10 × 10, so the logarithm is exactly 3, and this page confirms that by raising the base to the candidate in whole-number arithmetic instead of trusting a floating-point division. Many calculators take one logarithm and divide it by another, which is why they show a stray digit at the end. Here the near-miss is only a reason to test the exact answer, not an answer in itself.
What is the difference between log and ln?
Only the base. ln is the natural logarithm, taken in base e (about 2.718281828), the base that growth and decay are written in. log without a written base is the decimal logarithm, taken in base 10, in the notation used by engineering and by the sciences that count orders of magnitude. This page names both beside the base control, and any other positive base can be typed in its place. The rules on the page hold in every one of them.
Can I take the logarithm of a negative number?
Not as a real number, and the page refuses it instead of printing NaN. A positive base raised to any power is always positive, so no exponent takes a positive base to a negative argument. The same applies to zero: the base would have to be raised to an infinite power to reach it. Logarithms of negative numbers exist in the complex numbers, and that is outside this page.
What does solving for the base do?
It reads the same relation the other way. If log_x(1000) = 3, then the base x is the number whose cube is 1000, which is 1000^(1/3) = 10. This page answers that directly, so a question stated with an unknown base does not have to be rearranged by hand. The answer is checked the same way as every other: the base is raised back to the logarithm and the result must reproduce the argument exactly before the figure is called exact.
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