Formula
log_b(x) = y ⇔ bʸ = xlog_b(x) = ln(x) ÷ ln(b) (change of base)ln(x) = log_e(x), e ≈ 2.718281828 How it works
A logarithm answers the question “what power do I raise the base to, to get this number?” Because 10³ = 1000, log₁₀(1000) = 3. Logarithms undo exponents in the same way subtraction undoes addition.
Three bases come up constantly. Base 10 (the common log) is used for orders of magnitude, pH and decibels. Base e ≈ 2.71828 (the natural log, ln) appears in growth, decay and calculus. Base 2 is used in computing and information theory.
Any other base can be computed with the change-of-base formula: divide the natural log of the number by the natural log of the base. That is exactly what this calculator does, and the steps show the division so you can repeat it on a basic scientific calculator.
Frequently asked questions
Why can’t I take the log of zero or a negative number?
A positive base raised to any real power is always positive, so no real exponent can produce 0 or a negative number. Those logs are undefined in the real numbers.
What are the main log rules?
log(ab) = log a + log b, log(a/b) = log a − log b, and log(aⁿ) = n·log a. They hold in every base.
What is an antilog?
The antilog reverses a logarithm: the antilog of y in base b is bʸ. Use the exponent calculator to compute it.