\log

Logarithm

Function

In plain English

"How many times do I multiply the base to get this number?" $\log_{10} 1000 = 3$ ; $\log_2 8 = 3$ . Logs turn multiplication into addition ( $\log ab = \log a + \log b$ ), compress huge ranges into readable ones, and turn power laws into straight lines on a log-log plot. $-\log p$ is the surprise of an event with probability $p$ : rare events are very surprising. The base sets the units (base 2 gives bits, base e gives nats) and is often omitted in papers.

As code

Math.log10(1000);   // 3
Math.log2(8);       // 3
Math.log(Math.E);   // 1   (natural log, base e)
-Math.log2(1 / 8);  // 3 bits of surprise

See it

rate1.35

\log

is the inverse of the exponential: it counts multiplications. Equal ratios become equal distances, which is why log axes straighten power laws.

What it means, field by field

Mathematics

Inverse of exponentiation; log-log plots linearise power laws.

AI / ML

Log-likelihood and cross-entropy loss; log-scale axes for compute and parameters; log-probabilities (logits are unnormalised log-probabilities).

Physics

Boltzmann entropy S = k_B log W; decibels; magnitudes in astronomy.

Biology

pH (−log₁₀ of hydrogen-ion concentration); log fold-change in gene expression; sequence logos measured in bits.

In a research paper

Boltzmann’s entropy formula (engraved on his tombstone in Vienna): entropy as the log of the number of microstates W.

S = k_B \log W

Entropy grows with the number of microscopic arrangements consistent with what you observe, but only logarithmically.

Common misreadings

Assuming log means base 10. In ML and physics it is usually the natural log, but check.
Reading straight lines on log-log plots as proof of a power law. Many curves look straight over two decades.
Treating a small change in log loss as small. 0.1 nats is a substantial improvement at the frontier.

Connected symbols

e^{x}

Exponential

\prod

Product (capital pi)

\propto

Proportional to

\mathcal{L}

Loss function (calligraphic L)

\sum

Summation (capital sigma)

Where it appears on this site

∂ Shannon entropy ∂ Cross-entropy (the language-model loss)§ 1. Learning to Read Equations § 4. Probability, Entropy, and Information § 5. Complexity and Systems Thinking § 6. The Mathematics Inside Modern AI § 7. The Physics–Biology–AI Connection Lab