Log-Absolute Value

📐 Definition

For $x \in \mathbb{R}\setminus\{0\}$,

$$f(x) = \log |x|$$

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captures the logarithm of the magnitude while discarding sign.

Domain and Codomain

Defined for nonzero real inputs; outputs are real and handle wide dynamic ranges compactly.

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⚙️ Key Properties

$$\log|xy| = \log|x| + \log|y|$$

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$$\frac{d}{dx}\log|x| = \frac{1}{x}, \qquad x \ne 0$$

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$$\log|-x| = \log|x|$$

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🎯 Special Values and Limits

$x$	$\log\lvert x\rvert$
$1$	$0$
$-1$	$0$

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

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🔗 Related Functions

Absolute value $|x|$ supplies the magnitude and $\log(x)$ provides the sign-sensitive logarithm on $(0,\infty)$. Log-sum-exp is a separate stabilized log-domain primitive used for aggregating exponentials.

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Usage in Oakfield

Oakfield primarily uses log-of-magnitude ideas via stabilized forms rather than a raw log(|x|):

• Remainder operator: SIM_REMAINDER_NONLINEARITY_LOG_ABS evaluates log1p(|x|/\epsilon) to provide a safe “log magnitude” response for both small and large values.
• Thermostat soft clamps: smooth clamp logic uses log1p(exp(\cdot)), which is closely related to log-of-magnitude compression in practice (avoids hard clipping artifacts).
• Diagnostics: several statistics/metrics operate in log space (e.g. spectral entropy uses log over normalized power), though not always as log(|x|) specifically.

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Historical Foundations

📜 Logarithmic Scales

Logarithmic mappings of magnitude are a standard technique for representing wide dynamic ranges, appearing across scientific measurement and signal analysis.

🌍 Modern Perspective

$\log|x|$ is frequently used as a numerically stable way to compare multiplicative magnitudes additively, while intentionally discarding sign information.

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📚 References

• Bracewell, The Fourier Transform and Its Applications
• NIST Digital Library of Mathematical Functions (DLMF), §4