Log Calculator
Compute the logarithm of any number in any base, with natural log and a power check included.
How it works
log_b(x) = ln(x) ÷ ln(b)
A logarithm is an exponent in reverse: log base 10 of 1,000 is 3 because 10³ = 1,000. Any base converts to any other through the change-of-base rule — divide natural logs (or base-10 logs) — which is exactly how this calculator handles arbitrary bases.
Worked example
Log base 10 of 1,000: since 10 × 10 × 10 = 1,000, the answer is 3. By change of base, ln(1000) ÷ ln(10) = 6.9078 ÷ 2.3026 = 3.
Frequently asked questions
What’s the difference between log, ln and lg?
By convention log usually means base 10 (and base 2 in computing contexts), ln always means base e ≈ 2.71828, and lg most often means base 2. When it matters, the base is written as a subscript.
Why can’t I take the log of a negative number or zero?
A positive base raised to any real power stays positive, so no exponent ever produces 0 or a negative. The log of such numbers has no real value.
Where do logarithms show up in real life?
Any scale that grows by multiplication: decibels for sound, the Richter scale for earthquakes, pH for acidity, and doubling-time math for money and populations.
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This calculator is for educational purposes. Double-check important results before acting on them.