DIY Cost Tools
Logarithm Calculator
MATH CALCULATOR
Calculate a logarithm using any valid base, plus the natural logarithm and common base-10 logarithm.
How it works
A logarithm answers the question: “To what exponent must the base be raised to produce this value?” The change-of-base formula handles any valid base.
Formula
logᵦ(x) = ln(x) ÷ ln(b), for x > 0, b > 0, and b ≠ 1.
Example
log₁₀(1000) = 3 because 10³ = 1000.
Important limits
- Real logarithms require a positive input value.
- The base must be positive and cannot equal 1.
- Results are limited by browser floating-point precision.
Frequently asked questions
What is the natural logarithm?
It is the logarithm with base e, commonly written ln(x).
What is a common logarithm?
It uses base 10 and is often written log(x) or log₁₀(x).
Choosing a logarithm base
The base determines how the result should be interpreted. Base 10 is common for scientific scales, base 2 appears in computing, and the natural logarithm uses base e in growth and decay models. The input value must be greater than zero, while the base must be positive and cannot equal 1.
A logarithm answers the question: “What exponent produces this value?” For example, log base 10 of 1,000 is 3 because 10³ equals 1,000. If you compare results from another calculator, confirm that both tools use the same base and rounding precision. Very small or very large inputs may be easier to check by reversing the result with exponentiation.
Review notes
The calculation uses the standard logarithm change-of-base identity. Last reviewed September 30, 2026.