Negative Log Calculator
The negative logarithm — the transform behind pH, pKa, decibels of loss and the log-loss of a machine-learning model.
−log₁₀ x turns a small positive number into a manageable one, and reverses the direction: smaller x gives a larger result.
How the negative log calculator works
−log₁₀ x turns a small positive number into a manageable one, and reverses the direction: smaller x gives a larger result. That is exactly why pH works — a hydrogen-ion concentration of 10⁻⁷ becomes a pH of 7.
Because each unit is a factor of ten, differences on the scale are ratios in the original quantity: pH 3 is a thousand times more acidic than pH 6.
Formula: −log₁₀ x; x = 10^(−result)
Worked examples
| Inputs | Negative logarithm | Note |
|---|---|---|
| 10⁻⁷ — pure water | 7 | pH 7, neutral |
| 10⁻³ | 3 | pH 3, strongly acidic |
| Natural log-loss | 0.22314355 | 0.223 — the loss for a confident correct prediction |
FAQFrequently asked questions
Why is pH a negative logarithm?
Because hydrogen-ion concentrations are tiny numbers with many decimal places. Negating the log turns 0.0000001 into a clean 7.
What does one pH unit mean?
A tenfold change in concentration. pH 4 is ten times more acidic than pH 5, and a hundred times more than pH 6.
Can pH be negative?
Yes — concentrated strong acids can reach below zero, though measurement gets difficult there.
Where else does the negative log appear?
pKa in chemistry, decibels of attenuation, log-loss in machine learning, and the surprisal of an event in information theory.
Why does −log turn small numbers into large ones?
Because the log of a number below 1 is negative, so negating it flips the sign. The smaller the input, the larger the output.
Where these figures come from
- NIST Digital Library of Mathematical Functions — reference definitions for elementary and special functions
- Wolfram MathWorld — definitions and formulas for every topic on this page
- NIST/SEMATECH e-Handbook of Statistical Methods — the statistical formulas (mean, variance, z, confidence intervals, sample size)
- National curriculum in England — Mathematics — the terms and methods taught in UK schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.