Logarithm Calculator

A logarithm answers the question: to what power must I raise this base to get that number? log₁₀(1000) = 3 because 10³ = 1,000. Logarithms turn multiplication into addition, which is why they appear everywhere growth is measured.

Enter a positive number and pick a base — 10 for common logs, e for the natural log (ln), 2 for binary, or any custom base. All four common versions are shown together so you can compare them.

The formula

log_b(x) = ln(x) ÷ ln(b) — the power to which b must be raised to give x

Worked example

log₁₀(100) = 2, because 10² = 100. The same number in base 2 is about 6.644, and its natural log is about 4.605.

The example is pre-loaded in the result panel — press Calculate with your own numbers to replace it.

Frequently asked questions

Why is there no logarithm of zero or a negative number?

Because no power of a positive base ever reaches zero or goes negative — 10^x is always positive. The logarithm is therefore undefined (in real numbers) for values of zero or below.

Where are logarithms used in real life?

Earthquake magnitude (Richter), sound volume (decibels), acidity (pH), and the number of steps in binary search all use logarithmic scales, because the underlying quantities span enormous ranges.

How are logs and exponents related?

They are inverses: log_b(x) = y exactly when b^y = x. If you know 2^10 = 1,024, then log₂(1024) = 10.

Category: Math calculators

Logarithm in the selected base	2.0000
ln (base e)	4.6052
log₁₀	2.0000
log₂	6.6439