logarithm

Logarithm needs number to operate now: a logarithm is the exponent for a base and argument within the number system. Remove number right now and the logarithm concept ceases to function — it has no operands, no base, no value. The removal test passes at object level (Law 8): this is not the meta-level 'all concepts need number' shadow (Law 2b) but a genuine operational dependency of logarithm on the number system.

Pinned senses against both accepted carves (Law 11d): function's carve assigns each domain element to exactly one codomain element, with parameters (domain, codomain, assignment rule, uniqueness). logarithm's accepted carve 'assigns to each positive real number x the exponent to which a chosen base b must be raised to produce x' has domain the positive reals, codomain the reals, rule x -> log_b(x), and its own parameter (3) - 'the unique value y such that b^y = x' - states function's uniqueness constraint verbatim. Law 9 test: a competent speaker calls the logarithm a function - log_b is the canonical inverse-function correspondence. No nearer accepted kind exists on the board (no inverse-function entry), so function is the nearest rung; not a ladder leap.

Direction tested: DEPENDS_ON (remove Y — does X stop operating?). A slide rule operates by aligning logarithmic scales; without the concept of logarithms, the device has no mechanism and cannot function. This is a present-tense operational dependency, not historical.

The cepstrum (1960s, Bogert et al.) fundamentally uses the logarithm of the power spectrum as its first computational step: cepstrum = IFT(log(power_spectrum)). The logarithm function predates cepstrum by centuries and provides a core operation the cepstrum depends on.

The cepstrum algorithm operationally requires the logarithm: FFT → magnitude → natural log → inverse FT. Remove the logarithm and the cepstrum cannot operate — the quefrency domain emerges from the log of the power spectrum. This passes the removal test: without logarithm, there is no cepstrum.

Cepstral coefficients are computed via FFT → magnitude → natural logarithm → inverse FT. The logarithm is operationally necessary — remove it and the coefficients cannot be computed. The removal test passes: without logarithm, there is no cepstrum and no cepstral coefficients.

Cepstral analysis takes the logarithm of the magnitude spectrum as a core step. The mathematical concept of logarithm (1614, Napier) predates cepstral analysis (1963) by centuries. Which-came-first test passes.

The cepstrum-coefficient is computed as IDFT(log(|X[k]|²)). The logarithm concept (1600s, Napier/Bregg) predates cepstrum-coefficients (1963, Bogert). Logarithm is used as the mathematical machinery in the computation. Which-came-first test passes.

Homomorphic filtering operates by first taking the logarithm of the signal (converting convolution into addition), then FFT, then filtering in frequency domain, then IFFT, then exponential. The logarithmic transform is the indispensable first step — remove it and the technique cannot convert multiplicative convolution into additive operations. This is a present-tense operational dependency, not historical association.

Cepstral-envelope computation takes the logarithm of the spectrum before applying the inverse FFT. The logarithm is essential: without log-compression, the cepstrum cannot separate convolution from multiplication, which is the whole point of the quefrency domain. Logarithm predates and feeds into cepstral-envelope.

Law 7 which-came-first, from independent historical reasoning, not the graph epoch. The logarithm was invented by John Napier in 1614 (Canon Mirificus) as the computation-reducing function that turns products into sums; the bel was coined at Bell Telephone Laboratories in the 1920s as a named unit for logarithmic ratios in telephone transmission. The bel's entire operative content is the base-10 logarithm of a power ratio — L = log10(P1/P0) bels: the unit is the log10 value of the ratio given a name, and without the log10 function there is nothing to name. Pinned sense (Law 11d): the bel per its accepted definition as the whole-logarithm ratio unit, the logarithm per its accepted definition as the function log_b(x) = y iff b^y = x. Reverse fails: the logarithm predates and does not require the bel. The board already accepts this family's parallel lineage — exponential-function DERIVED_FROM logarithm, ruled on the same 'Napier's 1614 canon predates' reasoning — so direction and type are consistent with the accepted record.

Law 8b removal test, pinned sense: the neper as the natural-logarithm ratio unit whose value is L = ln(P1/P0). The neper's entire mechanism is applying the natural logarithm to a ratio — its value IS a logarithm. Remove the logarithm construct and there is no ln by which to compute the level from the ratio; the named unit has no operating content left (a plain ratio is exactly what its own definition distinguishes it from). The parallel decibel-DEPENDS_ON-logarithm claim stands on the identical test (its value is 10*log10 of the ratio). The log is the mechanism, not merely the origin: the origin is already recorded separately (neper DERIVED_FROM logarithm, accepted), mirroring image-frequency's dual accepted edges to heterodyning.

Law 8b removal: the decibel is a logarithmic unit expressing ratios of power or amplitude. Remove logarithm and the decibel ceases to operate — its definition is 10·log10(P1/P0) for power or 20·log10(V1/V0) for amplitude. The logarithm is the mathematical mechanism by which the decibel persists.

Law 7 which-came-first, from independent historical reasoning, not the graph epoch. The natural logarithm — the logarithm with base e — emerged from 17th-century analysis (the area under 1/x, the series for ln) and stood in the logarithmic tables long before any unit was named for it; the neper was adopted as the named natural-log unit of logarithmic transmission theory, the base-e counterpart of the bel. The neper's entire operative content is the natural logarithm of a ratio — L = ln(P1/P0) nepers: the unit is the ln value of the ratio given a name, and without the ln function there is nothing to name. Pinned sense (Law 11d): the neper per its accepted definition as the natural-logarithm ratio unit, the logarithm per its accepted definition as the function with base parameter, base e being its named natural-logarithm case. Reverse fails: the natural logarithm predates and does not require the neper.

log-sum-exp(x) = log(∑_i exp(x_i)) is defined using the logarithm. Remove the logarithm function and log-sum-exp has no definition — its very identity depends on the logarithm operation. The removal test passes: without logarithm, log-sum-exp ceases to operate as a function.

Which came first (Law 7), from independent historical reasoning: the logarithm was invented first — Napier's canon of 1614 predates the exponential function as a formal object by over a century. Euler's e^x notation (c. 1731) built the exponential as the inverse of the natural logarithm: exp(x) = y ⇔ ln(y) = x is the standard construction, and my definition names that inverse relation as one of its defining representations. The logarithm existed and fed into the exponential's formulation; the arrow points from the later object to the earlier source.