MATH · LOGARITHM
Logarithm Calculator
Calculate the natural log, log base 10, log base 2, or any custom base logarithm. Results to 6 significant figures with full domain validation.
2
log_10(100) = 2
About This Calculator
The logarithm answers the question "what exponent do I raise this base to, to get this value?" Logarithms appear throughout science, engineering, computing, music, and finance — from decibels to earthquake magnitudes to information theory.
How It Works
Select a base — natural (e), common (10), binary (2), or custom — and enter the value. The calculator uses the change-of-base formula for custom bases and JavaScript's native log functions for the standard bases. Results are given to 6 significant figures. The domain is strictly x > 0; x ≤ 0 and base = 1 are errors.
The Formula
log_b(x) = ln(x) / ln(b)
- x
- value (must be > 0)
- b
- base (must be > 0 and ≠ 1)
Frequently Asked Questions
- What is the natural logarithm?
- The natural logarithm (ln) uses Euler's number e ≈ 2.71828 as its base. It appears naturally in growth and decay equations, probability, and calculus.
- Why can't I take the log of zero or a negative number?
- Logarithm is the inverse of exponentiation. No real-number power of a positive base can produce zero or a negative result, so the logarithm is undefined for x ≤ 0. Attempting ln(0) produces negative infinity, which this calculator rejects as an error.
- Why is base 1 rejected?
- 1^x = 1 for all x, so there is no finite exponent that makes 1^x = y for y ≠ 1. log_1(x) is undefined.