Logarithm Calculator
Free logarithm calculator. Calculate log base 10, natural log (ln), log base 2, and any custom base logarithm with step-by-step explanation.
Free Tool
No Sign-up
Instant Results
Logarithm Calculator
log₁₀(x)
—
ln(x)
—
log₂(x)
—
Comprehensive Guide & Practice
Master the mathematical principles, formulas, and step-by-step methods with worked examples and FAQs.
1
1. What Is a Logarithm?
A logarithm answers the question: to what power must a base be raised to get a given number? Written as log_b(x) = n, it means bⁿ = x.
Example: log₁₀(1000) = 3 because 10³ = 1000. log₂(8) = 3 because 2³ = 8.
2
2. Types of Logarithms
- Common log (log₁₀): Base 10. Used in decibels (sound), pH (chemistry), Richter scale (earthquakes).
- Natural log (ln): Base e ≈ 2.71828. Used in calculus, compound interest, population growth, and physics.
- Binary log (log₂): Base 2. Used in computer science, information theory, and algorithm complexity (e.g. O(log n)).
3
3. Logarithm Rules
- Product: log(xy) = log(x) + log(y)
- Quotient: log(x/y) = log(x) − log(y)
- Power: log(xⁿ) = n × log(x)
- Change of base: log_b(x) = ln(x)/ln(b)
See also: Exponent Calculator | Scientific Notation