Log Calculator
Calculate the logarithm of a number at any base.
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About this calculator
How This Log Calculator Works
Enter a number and a base, and the calculator returns the logarithm of that number in the given base, along with the natural log (base e) and common log (base 10) for reference.
Worked Example
log₁₀(1000) = 3, since 10³ = 1000. The natural log of 1000 is 6.9078.
The Formula
Since most calculators only have built-in functions for natural log (ln) and log base 10, this tool uses the change of base formula: logᵦ(x) = ln(x) ÷ ln(b) — dividing the natural log of the number by the natural log of the desired base gives the logarithm in that base.
What a Logarithm Actually Means
logᵦ(x) answers the question "b raised to what power equals x?" It's the inverse operation of exponentiation — just as subtraction undoes addition, logarithms undo exponents. Base 10 (common log) shows up often in science and engineering (pH, decibels, the Richter scale); base e (natural log) is central to calculus and continuous growth/decay; base 2 is common in computer science, where it represents doublings. See the Exponent Calculator for the reverse operation.
Worked example
The log base 10 of 1,000 is 3, since 10³ = 1,000.
Frequently asked questions
What does a logarithm actually mean?
log_b(x) answers "what power do I raise b to, to get x?" It's the inverse operation of exponentiation, the same way division is the inverse of multiplication.
What's the difference between log, ln, and log10?
They're the same operation at different bases: "log10" (or often just "log") uses base 10, "ln" (natural log) uses base e ≈ 2.71828. This calculator computes any base you enter, plus ln and log10 for reference.
Why can't I take the log of a negative number or zero?
No real power of a positive base produces a negative result or zero, so the logarithm is undefined there — this is why the input must be a positive number.