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log_b(x) = ln x ÷ ln b

Logarithm calculator

Enter a value and a base. The page returns the logarithm, the natural and base-10 forms, the antilog and the change-of-base working.

Short answer: log₁₀(1000) = 3 because 10³ = 1000. The same value in base 2 is about 9.966, and in base e about 6.908 — the logarithm changes with the base, the number does not.

Reading a logarithm

A logarithm is an exponent, so small changes in its value mean large changes in the number. pH, the Richter scale and decibels are all logarithms, which is exactly why a change of one unit in any of them is a tenfold change in the underlying quantity.

Worked example

log₂(1024) = 10 because 2¹⁰ = 1024 — the reason a kilobyte was 1024 bytes. Changing the base to e gives ln(1024) ÷ ln(2) = 6.931 ÷ 0.693 = 10, the same answer through the change-of-base form.

Logarithm questions

What does a logarithm answer?

The exponent you need: log₂(8) = 3 because 2³ = 8. It reverses exponentiation, which is why it turns multiplication into addition and why graph scales use it.

How is a log in an unusual base computed?

By change of base: log_b(x) = ln(x) ÷ ln(b). The page prints that line with the actual values, so a homework answer can show the step rather than only the result.

Why is the value restricted to positive numbers?

Because no power of a positive base is negative or zero. log(−5) has no real answer, so the page refuses it instead of returning NaN.

What is the difference between log, ln and log₁₀?

ln is base e (2.718…), log₁₀ is base 10, and "log" depends on the field — base 10 in school, base 2 in computer science, base e in most mathematics. All three are shown here so the notation cannot mislead you.

Related tools

Scientific notation converterVery large and very small numbers.pH calculatorA concentration on a log scale.Quadratic equation calculatorRoots, discriminant and vertex.