How it works
A logarithm answers "what power do I raise the base to, to get this number?" log10(1000) = 3 because 103 = 1000.
The common log (base 10) is used for decibels, pH and the Richter scale. The natural log (base e ≈ 2.71828) appears in growth, decay and calculus. Base 2 counts doublings and bits.
Any base can be computed from natural logs with the change-of-base formula.
Formula
log_b(x) = y ⇔ b^y = x; log_b(x) = ln(x) ÷ ln(b)
Questions people ask
What is the difference between log and ln?
On most calculators "log" means base 10 and "ln" means base e. In many math and programming contexts, "log" alone means the natural log, so check the convention.
Why is the log of zero or a negative number undefined?
No real power of a positive base gives zero or a negative number, so there is no real answer.
What are the main log rules?
log(ab) = log a + log b; log(a/b) = log a − log b; log(aⁿ) = n·log a. These hold in any base.