What does the logarithm calculator do?

It returns log_b(n) — the exponent that base b must be raised to in order to produce n. It supports log₁₀, ln, log₂, and any custom positive base not equal to 1.

What formula does the calculator use?

It uses change-of-base: log_b(n) = ln(n) ÷ ln(b). Internally the tool computes natural logs and divides to give a logarithm in any base.

Why must the input be positive?

log_b(n) is undefined for n ≤ 0 in the real numbers. No real exponent of a positive base will produce zero or a negative result, so the tool rejects those inputs.

Why is a base of 1 not allowed?

Every power of 1 is 1, so log_1(n) is undefined for any n ≠ 1 and indeterminate when n = 1. The same is true for bases ≤ 0.

Does this tool store my numbers?

No. The calculation runs entirely in your browser; nothing is sent to a server or saved between visits.

Is the logarithm calculator free?

Yes. It is free to use with no signup, no account, and no usage limit.

Logarithm Calculator

Compute log₁₀, ln, log₂, or any custom base using change-of-base.

Written by Golam Rabbani, Founder & Lead Engineer

This interactive tool requires JavaScript. Read the formula and worked example below; you can compute the result by hand.

How to use this logarithm calculator

Pick a base: log₁₀, ln (base e), log₂, or Custom.
Enter the number (n) — must be greater than zero.
If you chose Custom, enter the base (positive and not equal to 1).
Press Calculate to see the logarithm.
Use Copy to put the result on your clipboard, or Reset to clear the inputs.

About this logarithm calculator

The logarithm calculator returns log_b(n) — the exponent to which a base b must be raised to produce a number n. It supports the three most common bases (10, e, 2) as quick presets, plus a custom base for any other positive value not equal to 1.

The formula uses change-of-base: log_b(n) = ln(n) ÷ ln(b). This lets a single internal natural-log function compute logarithms in any base. For example, log₁₀(1000) = 3 because 10^3 = 1000. ln(e) = 1 because e^1 = e (and e ≈ 2.71828). log₂(8) = 3 because 2^3 = 8. With a custom base of 5, log_5(125) = ln(125) ÷ ln(5) ≈ 4.8283 ÷ 1.6094 = 3. The calculator only accepts positive numbers for n because log_b(n) is undefined for zero or negative arguments in the real numbers.

As a second concrete example, audio engineers use decibels, defined as dB = 10 × log₁₀(P₂ ÷ P₁). If a speaker has 200 W output versus a reference of 1 W, enter base = 10, number = 200 and multiply the result by 10: log₁₀(200) ≈ 2.301, so the level is about 23.01 dB above reference. Logarithms appear in scientific scales (decibels, pH, Richter), information theory, algorithm analysis, and growth/decay modelling.

FAQ

What does the logarithm calculator do?: It returns log_b(n) — the exponent that base b must be raised to in order to produce n. It supports log₁₀, ln, log₂, and any custom positive base not equal to 1.
What formula does the calculator use?: It uses change-of-base: log_b(n) = ln(n) ÷ ln(b). Internally the tool computes natural logs and divides to give a logarithm in any base.
Why must the input be positive?: log_b(n) is undefined for n ≤ 0 in the real numbers. No real exponent of a positive base will produce zero or a negative result, so the tool rejects those inputs.
Why is a base of 1 not allowed?: Every power of 1 is 1, so log_1(n) is undefined for any n ≠ 1 and indeterminate when n = 1. The same is true for bases ≤ 0.
Does this tool store my numbers?: No. The calculation runs entirely in your browser; nothing is sent to a server or saved between visits.
Is the logarithm calculator free?: Yes. It is free to use with no signup, no account, and no usage limit.