A logarithm answers the question "to what power must base b be raised to equal x?" Formally, log_b(x) = y means b^y = x. For example, log_2(8) = 3 because 2^3 = 8.

What is the change-of-base formula?

The change-of-base formula lets you compute any logarithm using natural log (or log base 10): log_b(x) = ln(x) / ln(b). This tool shows this breakdown for every calculation.

ln is the natural logarithm — logarithm base e (≈ 2.71828). "log" without a stated base usually means log base 10 in engineering and science contexts, but log base 2 in computer science. This tool labels each base clearly.

Why is log of a negative number undefined?

Logarithms are only defined for positive real numbers (x > 0) and positive bases not equal to 1. A negative argument or base produces a complex number, which is outside the scope of this tool.

log_b(1) = 0 for any valid base b, because b^0 = 1.

What is log base 2 used for?

Log base 2 (binary logarithm) is widely used in computer science for measuring information in bits, analyzing algorithm complexity (e.g., binary search is O(log n)), and data compression.

Logarithm Calculator

Compute log base 2, log base 10, natural log (ln), or any custom base — with the change-of-base formula shown step by step.

Input

Number (argument x, must be > 0)

Base

Result

Enter a number and press Calculate

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Summary

The Logarithm Calculator evaluates log b(x) for any positive base and argument. It covers the three most common cases — log 2 (binary log), log 10 (common log), and ln (natural log) — plus an arbitrary base mode. Every result is accompanied by the change-of-base formula so you can see exactly how the answer was derived.

How it works

Enter the number (argument) you want to take the logarithm of.
Choose a base: 2, 10, e (natural log), or type a custom base.
Click Calculate — the result appears instantly using the built-in Math.log function.
The change-of-base breakdown (ln x / ln b) is shown below the result.

Use cases

Convert between binary, decimal, and natural logarithms.
Verify log calculations done by hand in math or engineering coursework.
Determine the number of bits required to represent a given number of values.
Calculate pH levels, decibels, or earthquake magnitude (Richter scale).
Solve exponential growth and decay problems in finance or science.
Check information entropy calculations in data compression.

Frequently Asked Questions

Last updated: 2026-05-28 · Reviewed by Nham Vu