The logarithm: the exponent in reverse
A logarithm answers the question what power do I raise the base to in order to get this number. This calculator finds the log of any positive number in any base, and also reports the three you meet most: natural (base e), common (base 10), and binary (base 2).
How it is computed
For any base, log base b of x = ln(x) / ln(b) — the change-of-base rule. That is how a calculator finds a base-7 or base-3 log from natural logs it already knows. The result is the exponent that turns the base back into x, which the check row confirms.
The three common bases
| Notation | Base | Used for |
|---|---|---|
| ln | e (2.718…) | growth, calculus |
| log | 10 | scientific notation, pH, decibels |
| lg or log₂ | 2 | computing, information |
Related math tools
To go the other way — raise a base to a power — use the exponent calculator.
Worked example
log base 10 of 1000 asks 10 to what power is 1000. Since 10 cubed is 1000, the answer is 3.
FAQ
Why must the number be positive?
No real power of a positive base ever produces zero or a negative number, so the logarithm of zero or a negative is undefined in the real numbers.
What bases are allowed?
Any positive base except 1. Base 1 fails because 1 to any power is always 1, so it can never reach another value.