A logarithm calculator computes log_b(x) — the power you'd have to raise base b to in order to get x. The Toolenza calculator handles base 10 (common log), base e (natural log, ln), base 2, and arbitrary user-chosen bases.

Why logarithms are everywhere

Logarithms turn multiplication into addition and exponentiation into multiplication. That's why they show up wherever quantities span many orders of magnitude:

• pH — log scale of hydrogen-ion concentration (base 10).
• Decibels — log scale of sound intensity ratios (base 10). +10 dB = 10× the power.
• Earthquake magnitude (Richter, moment) — log scale of energy release (base 10). Magnitude 6 releases ~32× the energy of magnitude 5.
• Star magnitude — log scale of brightness (with a 2.512 base by convention). Each magnitude step ≈ 2.5× brightness.
• Big-O notation — O(log n) algorithms scale logarithmically (binary search, balanced trees). 30 steps to search a billion items.
• Information theory — bits are log₂ of the number of distinguishable states.

The three bases that matter

• log = log₁₀ ("common log") — natural for human-scale measurements (pH, dB, decimal place values).
• ln = log_e ("natural log," base e ≈ 2.71828) — falls out of every calculus result; describes continuous growth and decay.
• lg = log₂ — natural for computer science and information theory.

Useful identities

• log(ab) = log(a) + log(b) — multiplication → addition.
• log(a/b) = log(a) − log(b) — division → subtraction.
• log(aⁿ) = n × log(a) — exponentiation → multiplication.
• log_b(x) = ln(x) / ln(b) — change-of-base formula; any base from any other.