Calculate logarithms with any base. Supports natural log (ln), common log (log₁₀), binary log (log₂), and custom base.
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Calculate logarithms instantly with our comprehensive calculator. Find ln (natural log), log10 (common log), log2 (binary log), or logarithm with any custom base. Perfect for mathematics, science, and engineering.
A logarithm is the inverse operation of exponentiation. log_b(x) = y means b^y = x. For example, log10(100) = 2 because 10^2 = 100. Common types include natural log (ln, base e ≈ 2.718), common log (log10), and binary log (log2). Logarithms are essential in science, engineering, and finance.
Logarithm Change of Base
Solve for time in compound interest and population models.
Sound intensity uses logarithmic scale (dB = 10×log10).
Calculate pH = -log10[H+] for acid-base chemistry.
Binary logarithms for algorithm complexity analysis.
Calculate natural, common, binary, or any base logarithm
Shows step-by-step calculations
Handles both integers and decimals
Uses the change of base formula for custom bases
Perfect for math, science, and engineering
ln (natural log) uses base e ≈ 2.71828, while log (common log) uses base 10. In some contexts, 'log' without a base can mean natural log, but in calculators and most applications, log means base 10.
Logarithm of zero is undefined because no power of a positive base equals zero. Logarithm of negative numbers isn't defined in real numbers because positive bases raised to any real power always give positive results.
The change of base formula allows you to calculate any logarithm using natural or common logs: log_b(x) = ln(x)/ln(b) = log(x)/log(b). This is how calculators compute logarithms with arbitrary bases.
Logarithms compress large ranges into manageable scales. Base-10 logs power the pH scale (acidity), the Richter scale (earthquakes), and decibels (sound). Natural logs (base e) model continuous growth and decay in finance, biology, and physics, while base-2 logs measure information in bits and the depth of binary algorithms in computer science.