Log Calculator
One card, three logarithm tools: natural log ln(x), common log log₁₀(x), and custom base logb(x). Each evaluates inside RapidRatio’s decimal-first helpers—see why readable decimals matter. For quick expressions, try the calculator on the home page.
3-in-1 Log Tools
Tool A: Natural log ln(x)
Tool B: Common log log₁₀(x)
Tool C: Custom base logb(x)
What is a logarithm?
A logarithm answers: “To what power must I raise the base to get x?” If by = x, then logb(x) = y. The base b must be positive and not equal to 1, and x must be positive in the real-number system.
- Natural log ln(x) uses base e (≈ 2.71828). Common in calculus, growth models, and science.
- Common log log₁₀(x) uses base 10. Common in pH, decibels, and orders of magnitude.
- Custom base lets you compute log2(8) = 3, log5(25) = 2, and similar problems from algebra class.
Change-of-base formula
Any base can be rewritten using natural logs: logb(x) = ln(x) / ln(b). Tool C computes logb(x) directly with the same high-precision path used across RapidRatio calculators.
Units, rounding, and limits
This calculator works with real positive numbers only—no complex logarithms. Displayed results use up to 14 decimal places with trailing zeros trimmed. Logarithms of 1 equal 0 for any valid base.
FAQ
What is the difference between ln and log?
On this page, ln means the natural logarithm (base e). log₁₀ or common log uses base 10. In some textbooks “log” without a subscript means base 10; in others it means base e—here Tool A is ln (base e) and Tool B is log₁₀.
Why is the logarithm of zero or a negative number undefined?
Real logarithms are defined only for positive inputs. No real number raised to a power yields zero or a negative result in the standard real-number definition, so the calculator reports an error instead of a misleading value.
How do I calculate a logarithm with a custom base?
Use Tool C: enter the positive number x and a positive base b that is not 1. The result is logb(x)—the exponent you raise b to in order to get x.