Logarithm Calculator

How to Use This Calculator

Enter the value (x) you want to take the logarithm of, and the base. Leave the base as 10 for common logarithm (log₁₀), enter "e" (≈ 2.71828) for natural logarithm (ln), or enter 2 for binary logarithm (log₂). The calculator uses the change-of-base formula internally.

Formula

The Three Most Common Logarithms

• Common logarithm log₁₀(x): Used in chemistry (pH = −log[H⁺]), decibels (dB = 10 log(P/P₀)), and earthquake magnitude (Richter scale). log₁₀(1) = 0, log₁₀(10) = 1, log₁₀(100) = 2, log₁₀(1000) = 3.
• Natural logarithm ln(x): Uses base e ≈ 2.71828. Fundamental in calculus (the derivative of ln(x) is 1/x), continuous compound interest (A = Pe^(rt)), and probability distributions. ln(1) = 0, ln(e) = 1, ln(e²) = 2.
• Binary logarithm log₂(x): Central to computer science — log₂(n) gives the number of binary digits needed to represent n values. log₂(8) = 3 because 2³ = 8. The time complexity of binary search is O(log₂ n).

Key Logarithm Properties

These properties allow simplification of complex logarithmic expressions:

• Product rule: log_b(xy) = log_b(x) + log_b(y)
• Quotient rule: log_b(x/y) = log_b(x) − log_b(y)
• Power rule: log_b(xⁿ) = n × log_b(x)
• Change of base: log_b(x) = log_a(x) / log_a(b) for any base a
• Inverse property: b^(log_b(x)) = x and log_b(b^x) = x

Real-World Applications

Logarithms compress large ranges of numbers into manageable scales. pH (acidity) uses log₁₀: each unit change represents a 10× change in hydrogen ion concentration. Sound intensity (decibels) uses log₁₀: a 10 dB increase means 10× more power. Earthquake magnitude (Richter) uses log₁₀: a magnitude 6 quake releases 32× more energy than a magnitude 5. Compound interest and population growth use ln. In data science, the log transform is frequently applied to skewed data before modeling to make distributions more symmetric.