Logarithm Calculator
Calculate logarithms in any base — common log (base 10), natural log (base e), binary log (base 2), or any custom base using the change-of-base formula.
Results
What is it?
A logarithm answers the question: "To what power must the base be raised to get x?" If bⁿ = x, then logᵦ(x) = n. The three most common bases are 10 (common log, used in pH and decibels), e ≈ 2.71828 (natural log, used in calculus and growth models), and 2 (binary log, used in computer science and information theory).
How to use
Enter a positive value for x and select the desired base. Choose "Custom" and fill in the Custom Base field to compute any arbitrary base using the change-of-base formula. All four reference logs (base 10, e, 2, and custom) are shown simultaneously.
Example scenario
log₁₀(100) = 2 because 10² = 100. ln(e³) ≈ 3. log₂(1024) = 10 because 2¹⁰ = 1024 (one kilobyte). For pH: pH = −log₁₀[H⁺], so [H⁺] = 10⁻⁷ gives pH = 7 (neutral).
Pro tip
Key logarithm laws: log(ab) = log(a) + log(b); log(a/b) = log(a) − log(b); log(aⁿ) = n·log(a). Change-of-base formula: logᵦ(x) = ln(x)/ln(b). Logarithms appear in the Richter scale (earthquakes), decibel scale (sound), compound interest formulas, and entropy calculations in information theory.