log_b(x) = y means b^y = x. It is the inverse of exponentiation. Example: log₂(8) = 3 because 2³ = 8.
Logarithm Calculator
Calculate natural logarithms (ln), decimal (log₁₀) and any custom base.
Natural, decimal and arbitrary-base logarithms with properties.
The logarithm log_b(x) is the exponent to which base b must be raised to produce x. It is equivalent to solving b^y = x. The natural logarithm (ln) uses base e ≈ 2.71828; the decimal (log₁₀) uses base 10; any other base can be entered. The tool uses the change-of-base formula: log_b(x) = ln(x) / ln(b).
Choose a base and enter the value of x.
- Select the base: natural (e), decimal (10) or custom.
- If you choose a custom base, enter it (> 0 and ≠ 1).
- Enter x > 0 and see the result with equation and properties.
Sources and references for this tool
These references help contextualize formulas, standards, APIs and limitations used on this page. They do not replace professional validation when a result has legal, financial, medical or operational impact.
- Logarithm — Wolfram MathWorldWolfram MathWorld — Definition, properties and identities of logarithms.
- Logarithm — WikipediaWikipedia — History, definitions and applications of logarithms in science and engineering.
- Logarithms — Khan AcademyKhan Academy — Introduction and exercises on logarithms with change of base.