Logarithm

📐 Definition

For $x > 0$ , the (natural) logarithm can be defined by the integral

\log x = \int_1^{x} \frac{dt}{t}

the inverse of the natural exponential, converting multiplicative changes into additive ones.

Domain and Codomain

Defined for positive real inputs; outputs are real and span the entire real line.

On $\mathbb{C}\setminus(-\infty,0]$ , the principal complex logarithm is $\operatorname{Log}(z)=\ln|z|+i\operatorname{Arg}(z)$ .

⚙️ Key Properties

Product, Quotient, and Power Rules

\log(xy) = \log x + \log y, \qquad \log\!\left(\frac{x}{y}\right) = \log x - \log y

\log(x^a) = a\,\log x

for $x,y>0$ (and with appropriate branch choices in the complex plane).

Derivative

\frac{d}{dx}\log x = \frac{1}{x}

Concavity

On $(0,\infty)$ , $\log x$ is concave, a key fact behind many inequalities and stabilization tricks.

🎯 Special Values and Limits

$x$	$\log x$
$1$	$0$
$e$	$1$

As $x \to 0^+$ , $\log x \to -\infty$ , and as $x \to \infty$ , $\log x \to \infty$ .

The logarithm is the inverse of the exponential on $(0,\infty)$ . Stabilized constructions such as log-sum-exp and log-absolute-value build on the same multiplicative-to-additive conversion.

Usage in Oakfield

Oakfield uses log/log1p primarily for stable measurements and smooth clamps:

Remainder nonlinearity: the remainder operator’s LOG_ABS mode uses log1p(|x|/\epsilon) to compress magnitudes without singularities at zero.
Spectral entropy diagnostics: runtime field statistics compute normalized spectral entropy using terms like $-\sum p\log p$ (with a log(n) normalization).
Smooth clamping / softplus: the thermostat operator uses a softplus-style expression log1p(exp(kx)) as part of its smooth clamp behavior.
Random sampling internals: some RNG paths use log in standard transforms (e.g. Box–Muller) when generating Gaussian deviates for stimuli/noise.

Historical Foundations

📜 Logarithms for Computation

Logarithms were introduced to simplify multiplication and division into addition and subtraction, a major practical advance for astronomy, navigation, and early scientific computation.

🌍 Modern Perspective

In modern analysis, $\log$ is treated as the inverse of $\exp$ with a canonical branch structure in the complex plane, making branch cuts and arguments central to correct usage.

📚 References

NIST Digital Library of Mathematical Functions (DLMF), §4
Ahlfors, Complex Analysis