The logarithm log_b(x) is the exponent you must raise b to in order to get x. For example, log₁₀(1000) = 3 because 10³ = 1000.

What is the difference between log and ln?

"log" typically refers to the common logarithm (base 10), used in science and engineering. "ln" is the natural logarithm (base e ≈ 2.71828), used in calculus and continuous growth models.

What is the change-of-base formula?

log_b(x) = ln(x) / ln(b) = log(x) / log(b). It lets you compute any base logarithm using the natural log (or any other base your calculator supports).

Why can't the base be 0 or 1?

Base 0 is undefined (0^n = 0 for all n > 0, so it can never equal x). Base 1 is undefined (1^n = 1 always, so it can only "equal" x when x = 1, making the result indeterminate).

Why must x be positive?

Logarithms are only defined for positive real numbers. There is no real exponent n such that b^n = 0 or b^n = negative number.

What is log base 2 used for?

Binary logarithm (log₂) appears throughout computer science: calculating the number of bits needed to represent n values, measuring information entropy, analyzing divide-and-conquer algorithm complexity.

Logarithm Calculator

Calculate log base 10, natural log (ln), binary log, or any custom base instantly — with step-by-step working.

Calculate Logarithm

Number (x) — must be positive

Base

Step-by-Step

Enter a number and select a base to see the working.

Common Logarithm Values

x	log₁₀(x)	ln(x)	log₂(x)

Copied!

Summary

This logarithm calculator computes log(x) for any base you choose — common logarithm (base 10), natural logarithm (base e), binary logarithm (base 2), or any positive custom base. Results are shown instantly with a clear step-by-step explanation using the change-of-base formula. All computation runs in your browser with no data sent to any server.

How it works

Enter the number (x) you want to find the logarithm of.
Select a preset base — log₁₀, ln (base e), log₂ — or type any custom positive base.
The result appears instantly as you type.
The step-by-step panel shows the change-of-base formula applied to your inputs.
Use the inverse field to verify: raise the base to the result and confirm you recover x.

Use cases

Solve algebra and pre-calculus homework involving logarithms.
Convert between logarithm bases using the change-of-base formula.
Compute decibel levels or Richter magnitude (base-10 log).
Work with information theory and entropy calculations (base-2 log).
Check natural-log values for calculus derivatives and integrals.
Verify logarithm identities: log_b(x*y) = log_b(x) + log_b(y).
Find exponents: if b^n = x, compute n = log_b(x).
Science and engineering: pH, stellar magnitude, bit depth analysis.

Frequently Asked Questions

Last updated: 2026-06-04 · Reviewed by Nham Vu