Math & Stats

Logarithm Calculator

Compute the logarithm of any value in any base with this free Logarithm Calculator. Quick buttons for base 10, base e, and base 2 included.

Logarithm Calculator

Value (x)

Base (b)

Method

How this calculator works

log_b(x) = ln(x) / ln(b) = Math.log(x) / Math.log(b). This change-of-base formula lets you compute a logarithm in any base b from the natural logarithm. The value x must be positive, and the base b must be positive and not equal to 1.

Enter the value (x) you want the logarithm of; it must be greater than 0.
Enter the base (b), or tap a quick button for base 10, e, or 2.
Click Calculate or edit a field to recompute automatically.
The tool applies the change-of-base formula log_b(x) = ln(x) / ln(b).
Read the result and the full expression such as log₁₀(1000) = 3.

Examples

Worked examples

Real numbers, end-to-end results.

log₁₀(1000)

log₁₀(1000) = 3

10 raised to the power 3 equals 1000.

log₂(8)

log₂(8) = 3

2 raised to the power 3 equals 8.

ln(e) with base e

logₑ(2.71828) = 1

The natural log of e is exactly 1.

Use cases

When to use it

Working with pH, decibels, and the Richter scale in science.
Solving exponential growth and decay equations in algebra.
Analyzing algorithm complexity such as log-base-2 running times.
Converting between logarithmic scales in engineering and finance.

FAQ

Frequently asked questions

What is a logarithm?

A logarithm answers the question: to what power must the base be raised to produce a given value? For example, log₁₀(1000) = 3 because 10³ = 1000.

What is the difference between log and ln?

log usually means the common logarithm with base 10, while ln is the natural logarithm with base e ≈ 2.71828. Both are just logarithms with different bases.

Why must the value be positive?

Logarithms are only defined for positive real numbers. There is no real power of a positive base that produces zero or a negative number, so the input must be greater than 0.

Why can't the base be 1?

A base of 1 raised to any power is always 1, so it can never equal any other value. Bases must be positive and different from 1 for the logarithm to be defined.

How does the change-of-base formula work?

Any logarithm can be computed as log_b(x) = ln(x) / ln(b), or equivalently log(x) / log(b). This calculator uses that formula to support any valid base.