Deep Dive: Exponentials and Logarithmic Graphs

#CODEX deep-dive: Exponentials and Logarithmic Graphs (WMA13, spec 3.1–3.3)

Provenance (2026-09-29). Read/grepped in full for this pass: research/veridian/WMA13-verified-facts.md (§1 spec text, §3 formula-booklet contents, §5.3 "Exponentials and logarithms", §6–8, §9 lesson-audit entries for this lesson); content/veridian/curriculum/wma13/exponentials-and-logarithmic-graphs.ts in full (1203 lines — mechanisms, both worked chains, the chain-drill, the marked solution, all seven trap-taxonomy items, the beyond-spec block, all six MCQs, the reference card, all 21 flashcards); content/veridian/technique-hub/wma13/codex-formula-and-method-bank.md, forge-exam-technique-guide.md and signal-frequency-and-boundaries.md in full; content/veridian/question-bank/wma13.ts (grepped — its only exponential item is a spec-4.4 differentiation question, out of this document's scope). No new primary source (PDF) was independently downloaded for this pass; every claim below is either a synthesis of facts already verified in WMA13-verified-facts.md, or an explicitly labelled VERIDIAN-original illustration. This document does not reproduce any of the lesson's worked chains, chain-drill, marked solution, MCQs or flashcards — see §7 for exactly what is additive here versus what already lives in those files.

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#1. Scope

Per the spec's own numbering (Specification Issue 3, April 2019, Unit P3, p.20, §3 "Exponential and logarithms"), items 3.1–3.3 name seven distinct sub-cases between them: 3.1 covers the graph of eˣ itself, together with the transformed form y = e^(ax+b) + c named in its own guidance note; 3.2 covers the graph of ln x as the inverse of eˣ, together with solving equations of the two named shapes, e^(ax+b) = p and ln(ax+b) = q ("is expected," the spec's own phrasing); 3.3 covers using a logarithmic graph to estimate the constants in y = axⁿ and, separately, in y = kbˣ, with the spec's own guidance stating outright which axis gets logged and what gradient and intercept each recover for each of the two forms. This document is organised around exactly those seven sub-cases, no more and no fewer: (1) the graph of eˣ, (2) the transformed graph y = e^(ax+b) + c, (3) the graph of ln x as the inverse of eˣ, (4) solving e^(ax+b) = p, (5) solving ln(ax+b) = q, (6) log-linearising y = axⁿ, (7) log-linearising y = kbˣ. Spec 4.4 (exponential growth/decay, differentiation of aˣ) is a sibling topic with its own lesson (exponential-growth-and-decay.ts) and its own verified-facts entries; it is deliberately excluded here even where a real question mixes both spec items in different parts (noted explicitly where it matters, §5.1 and §6).

#2. What's given, what must be memorised (this sub-topic only)

Verified directly against the formula booklet's own P3 section and the spec's own "Notation and formulae" list (WMA13-verified-facts.md §3):

• Printed in the booklet, P3-specific: $e^{x\ln a} = a^x$. Nothing else from 3.1–3.3 is printed.
• Carried forward from the P2 booklet section (so still "given," not memorised, for a P3 candidate): the change-of-base formula, $\log_a x = \dfrac{\log_b x}{\log_b a}$.
• Not printed anywhere — must be memorised: the three log laws themselves (product, quotient, power) are not in the spec's "must memorise" list by name, but nothing in the booklet states them either; a P3 candidate is expected to already know them from P1/P2, per the spec's own prerequisite line ("A knowledge of the specifications for P1 and P2... is assumed and may be tested"). This is the reason the lesson's own second teach block exists as a from-scratch recap rather than a pointer — there is genuinely no P3-side formula-sheet support for the single most-used identity in this topic, the power law.
• Two adjacent facts belong to spec 4.4, not here, and are flagged only so they aren't confused with anything above: $\dfrac{d}{dx}(a^x) = a^x\ln a$ must be memorised (not printed), and it is the booklet-given identity $e^{x\ln a}=a^x$ that makes it derivable on demand by rewriting $a^x$ as $e^{x\ln a}$ and applying the (also memorised) $e^{kx}\to ke^{kx}$ result. Spec 3.1–3.3 never needs differentiation at all; this identity's only job inside this topic is occasionally rewriting a base-$a$ exponential in base $e$ or vice versa before applying the log-linearisation mechanism in §3.6.

#3. Theory: the seven sub-cases in full

#3.1 The function eˣ

$e^x$ is defined for every real $x$, is strictly positive throughout (never zero, never negative), and is strictly increasing everywhere: for $a<b$, $e^a<e^b$, with no turning point and no point of inflection change of monotonicity. As $x\to-\infty$, $e^x\to0^+$ without ever reaching it (a horizontal asymptote at $y=0$); as $x\to+\infty$, $e^x\to\infty$ with no asymptote on that side. $e^0=1$ is the one point worth having by reflex, since it anchors both this graph and its inverse (§3.3). None of this is spec-3.1 content that needs proving from calculus — it is the graph's shape, read off directly, and the strict-increase property is the one fact the rest of this section leans on repeatedly (§3.3's inverse-existence argument, specifically).

#3.2 The transformed graph $y = e^{ax+b} + c$

Spec 3.1's own guidance names this transformed form explicitly, so a bare "$e^x$" sketch is not actually what gets examined — the transformed version is. Nothing new is needed beyond ordinary function-transformation reasoning applied to this one function: $c$ translates the whole curve vertically, carrying its asymptote from $y=0$ to $y=c$ (the term $e^{ax+b}$ itself still tends to $0$, so the curve still approaches a horizontal line, just a shifted one — this is a real distinction worth stating plainly, since a floor/ceiling constant like this $c$ is exactly what §5.3's domain-restriction questions turn on); $a$ and $b$ inside the exponent stretch and translate the curve horizontally, in the same combined way any $f(ax+b)$ does. The sign of $a$ matters for the shape (a negative $a$ flips increasing to decreasing, e.g. a decay curve $c+de^{-kt}$), but the mechanics of finding the asymptote and the intercept never change.

#3.3 ln x as the inverse of eˣ

An inverse exists as a genuine function only where the original is one-one (a many-one function would need its inverse to return two outputs for one input). §3.1 already established $e^x$ is strictly increasing on the whole of $\mathbb{R}$, so it is one-one everywhere with no domain restriction needed first — unlike a quadratic, which needs a restricted domain before it can be inverted at all. So $e^x$ has an inverse defined on all of $\mathbb{R}$, named $\ln x$ rather than left as "$e^x$ backwards." Domain and range swap directly from $e^x$'s own: domain of $\ln x$ is $\{x:x>0\}$ (the range of $e^x$), range is all of $\mathbb{R}$ (the domain of $e^x$). Graphically, swapping the roles of input and output is exactly reflecting every point $(p,q)$ to $(q,p)$, which is reflection in the line $y=x$ — so $y=\ln x$ is the mirror image of $y=e^x$ in $y=x$, a geometric fact true of every inverse-function pair, not a special property of this one. The defining cancellations follow: $\ln(e^x)=x$ for every real $x$; $e^{\ln x}=x$ for every $x>0$.

#3.4–3.5 Solving $e^{ax+b}=p$ and $\ln(ax+b)=q$

Both are the same single move, run in opposite directions: apply whichever function undoes the one already sitting in the equation. For $e^{ax+b}=p$: apply $\ln$ to both sides; the left side collapses to $ax+b$ by the cancellation above, leaving the linear equation $ax+b=\ln p$. For $\ln(ax+b)=q$: apply $e^{(\cdot)}$ to both sides instead; the left side collapses to $ax+b$, leaving $ax+b=e^q$. There is no second genuine method for either — WMA13-verified-facts.md documents no real worked example using any other route, which is exactly why the lesson deliberately carries no method-comparison block here (its own header states this reasoning). The one real subtlety worth naming as theory rather than as a trap: $p$ must be strictly positive for $e^{ax+b}=p$ to have any solution at all ($e^{ax+b}>0$ always), and this is not a special case to memorise separately — it is the domain fact from §3.1 read backwards, and it is exactly the mechanism §5.3 below is built on.

#3.6 Log-linearisation: two distinct sub-cases

A relationship $y=ax^n$ or $y=kb^x$, with the exponent or power unknown, cannot be read directly off a plot of $y$ against $x$ — the curve's shape depends on the unknown in a way that isn't identifiable by eye. Taking logs of both sides turns each into a straight line, but which variable needs logging differs between the two forms, because the unknown sits in a different place in each:

• $y=ax^n$ ($n$ unknown, sitting as a power of $x$): $\log_c y = \log_c a + n\log_c x$ for any valid base $c$, by the product law then the power law. Plotting $\log_c y$ against $\log_c x$ (both axes logged) gives a straight line with gradient $n$ and intercept $\log_c a$.
• $y=kb^x$ ($b$ unknown, sitting as the base of an exponent whose power is $x$ itself): $\log_c y = \log_c k + x\log_c b$. Plotting $\log_c y$ against unlogged $x$ gives a straight line with gradient $\log_c b$ and intercept $\log_c k$.

The rule for which axes to log is therefore not a fact to memorise in isolation — it falls straight out of asking "is the unknown a fixed power applied to $x$ (log both), or is $x$ itself the varying exponent (log only $y$)?" This is the one piece of theory the existing lesson states as a fact (self-explain prompt, reference card) without deriving the general principle behind it as its own named idea; stating it this way — as a diagnostic question to ask of any new relationship, not a memorised pair of recipes — is this deep-dive's own framing, not a restatement.

Base-dependence, precisely. The gradient of a $y=ax^n$ log-log line is $n$ regardless of which base $c$ the logs are taken in — $n$ was already an ordinary number once the power law pulled it out of the exponent, and that number doesn't change with $c$. The intercept of either line-type, though, is $\log_c(\text{constant})$ — a genuinely different number for a different $c$ — and so is the gradient of the semi-log $y=kb^x$ line specifically ($\log_c b$ changes with $c$, unlike $n$ above, because $b$ itself is what's being logged there, not pulled out by the power law). Un-logging any intercept, or the semi-log gradient, always means raising the same base the graph was actually plotted in — never base 10 by default habit. This three-way distinction (gradient of a power-law fit is base-invariant; gradient of an exponential fit is not; no intercept ever is) is the single most exam-relevant piece of theory in the whole sub-topic, because it is the exact fact the one located base-confusion error (§5.6) turns on.

Reading a fitted equation two different ways. Once $\log_c y = \log_c a + n\log_c x$ (or the semi-log equivalent) is established from two data points, there are genuinely two different tasks a question can set, not one: (i) read $n$ and $a$ (or $b$ and $k$) off as two separate numbers, each un-logged on its own — the ordinary case; or (ii) given the fitted linear log-equation directly, eliminate the logs algebraically in one continuous derivation to state the original relationship with no logarithm left in it anywhere — a task that needs the product and power laws run in the reverse direction from how they were used to linearise in the first place ($\log_c y = P + Q\log_c x \Rightarrow y = c^P \cdot x^Q$, by exponentiating both sides and splitting the right-hand exponent as a sum). These are not the same skill wearing different numbers — (ii) is a full derivation with its own method-mark structure, confirmed in the real record to be where marks are lost even by candidates who handled (i) correctly (§5.4).

Interpreting a fitted constant without further calculation. A distinct third task, once $b$ (or $n$) is already found: state in words what the constant means for the model, with no arithmetic at all beyond what's already been done. For $y=kb^x$: since $y(x+1)=kb^{x+1}=b\cdot kb^x=b\cdot y(x)$, the constant $b$ is the multiplier applied once per unit of $x$ — $b>1$ means a percentage increase of $(b-1)\times100\%$ per unit; $0<b<1$ means a percentage decrease of $(1-b)\times100\%$ per unit. Both directions are the same underlying fact, $b$ as a one-step multiplier, read differently depending only on which side of $1$ it falls.

#4. Short worked micro-examples

These are VERIDIAN-original, illustrative only — not reproductions of any past paper, and deliberately shorter than the lesson's own worked chains (one to two lines each, one sub-case at a time, no mark scheme attached). Every number below was computed directly for this pass.

(a) Solving $e^{ax+b}=p$. $e^{3x-2}=40 \Rightarrow 3x-2=\ln 40 \Rightarrow x=\dfrac{\ln 40+2}{3}\approx1.90$ (3 s.f.).

(b) Solving $\ln(ax+b)=q$. $\ln(5x+1)=2.4 \Rightarrow 5x+1=e^{2.4} \Rightarrow x=\dfrac{e^{2.4}-1}{5}\approx2.00$ (3 s.f.).

(c) Log-log fit, $y=ax^n$. A log-log plot passes through $(\log_{10}x,\log_{10}y)=(0.3,1.1)$ and $(1.1,3.5)$. Gradient $=\dfrac{3.5-1.1}{1.1-0.3}=\dfrac{2.4}{0.8}=3=n$. Intercept $=1.1-3(0.3)=0.2=\log_{10}a$, so $a=10^{0.2}\approx1.58$ (3 s.f.). Model: $y\approx1.58x^3$.

(d) Semi-log fit, $y=kb^x$. A semi-log plot ($\log_{10}y$ against unlogged $x$) passes through $(0,0.8)$ and $(5,2.05)$. Gradient $=\dfrac{2.05-0.8}{5}=0.25=\log_{10}b$, so $b=10^{0.25}\approx1.78$ (3 s.f.). Intercept $=0.8=\log_{10}k$, so $k=10^{0.8}\approx6.31$ (3 s.f.). Model: $y\approx6.31(1.78)^x$.

(e) Eliminating the logs entirely. Given directly $\log_{10}T=3-0.5\log_{10}x$: exponentiate both sides, $T=10^{3-0.5\log_{10}x}=10^3\times10^{-0.5\log_{10}x}=1000\times x^{-0.5}=\dfrac{1000}{\sqrt x}$ — no separate "read $a$, read $n$" step at all, the whole relationship in one line.

(f) Interpreting $b<1$ in words (the decaying case the real Jun 2026 Q5(b) does not itself cover, since its own $b\approx1.19>1$ — see §5.5 and §6). A model fits $b=0.85$. Since each unit-step multiplies the modelled quantity by $0.85$, the constant means the quantity is decreasing by 15% per unit ($(1-0.85)\times100=15$) — the direction word flips from "increasing" to "decreasing," and the percentage is $1-b$, not $b-1$, once $b$ is below $1$.

(g) Domain restriction. A model $T=15+6e^{-0.4t}$ has a genuine floor at $T=15$ (as $t\to\infty$, $e^{-0.4t}\to0^+$ but never reaches it, so $T\to15^+$ and never lower). Asked to find $t$ when $T=10$: rearranging gives $6e^{-0.4t}=-5$, which has no solution because $e^{-0.4t}>0$ for every real $t$, so $6e^{-0.4t}$ can never be negative — the equation breaks down before any log is even taken, not because of a rounding or algebra slip.

#5. Synthesis: every real exam angle this sub-topic has actually been tested from

Grounded in WMA13-verified-facts.md §3, §5.3, §6–9 and the counts independently tabulated in signal-frequency-and-boundaries.md. Series checked for this paper across the whole research pass: 8 read in full (Oct 2020, Oct 2021, Jan 2022, Jun 2022, Oct 2022, Jun 2023, Jan 2024, Jan 2025) plus one further series added by a later citation-currency audit, Jun 2026 (mark scheme and question paper only — no examiner report exists yet for that series, checked and confirmed 404 at the mirror used). That is 9 series checked in total; counts below are stated against that total and named explicitly, never as a percentage.

5.1 Equation-solving (§3.4–3.5) embedded inside a real-world modelling story is the dominant angle. signal-frequency-and-boundaries.md's own tally: a real-world exponential/log modelling question was located in 7 of the 8 fully-reviewed series (every one except Oct 2020), plus Jun 2026 — 8 of the 9 series checked. The overwhelming majority of these questions wrap solving $e^{ax+b}=p$ or (less often) $\ln(ax+b)=q$ inside a physical story (decay, growth, population) rather than testing the algebra bare — confirmed contexts include a gold-mine extraction decay model (Oct 2021 Q3), a tree mass/radius model (Oct 2021 Q7), an antibiotic-concentration decay model (Jan 2022 Q8), pond-area growth models (Jun 2022 Q4, Jan 2025 Q2), a charity-funding growth model (Jun 2023 Q7), a population/decay comparison model (Jan 2024 Q3), two temperature models (Jan 2024 Q5, Jan 2025 Q5) and a household-recycling growth model (Jun 2026 Q5). Scope caveat, stated plainly rather than left implicit: several of these same questions' other parts (the ones cited above for a rate-of-change or sign-drop error) are spec-4.4 differentiation, not this sub-topic — the equation-solving part of each is this sub-topic's own contribution to the question, the differentiation part belongs to the sibling lesson. No claim is made here about how many marks within each question are 3.1–3.3 specifically versus 4.4, because the verified-facts entries for several of these (Oct 2021 Q3/Q7, Jan 2024 Q3) do not break the mark allocation down to that level of detail.

5.2 Log-linear-graph parameter estimation as its own standalone question (§3.6) is real but comparatively rare. Confirmed in exactly 2 of the 9 series checked — Jun 2023 Q2 and Jun 2026 Q5 — each worth 6–7 marks and, unusually for this sub-topic, presented with no real-world story at all in the Jun 2023 case (bare $T$ and $x$, a log-log graph in base 6) and only a thin one in the Jun 2026 case. This is worth stating as its own finding: spec 3.3's headline technique (the one the spec's own guidance describes in the most mechanical, formulaic terms of the three sub-sections) is tested far less often as a dedicated question than 3.2's equation-solving is as an embedded skill — 2 series versus 8.

5.3 Domain-restriction / "no solution" reasoning (§3.4, the $p\le0$ or floor/ceiling case). Confirmed in exactly 1 of the 9 series — Jan 2024 Q5(d) — a genuine "explain why $T=5$ has no solution" question on $T=10+8e^{-Bt}$. The real mark scheme is precise about why the explanation must be phrased the way it is: credited wordings are "which is not possible," "cannot be done," or "you cannot find the log of a negative number"; explicitly not credited are "logs cannot be negative" or "you cannot have a negative time," both named in the scheme itself as "ambiguous/incorrect statements." This is a single-series finding, not a "confirmed pattern" — honestly stated as such, matching this course's own convention for distinguishing a well-attested trap from a real-but-so-far-singular one.

5.4 Algebraic elimination of logs entirely, distinct from reading two separate constants (§3.6). Confirmed in exactly 1 of the 9 series — Jun 2023 Q2(b) — asking for "an equation, not involving logs, linking $T$ with $x$" directly from a given $\log_6T=4-2\log_6x$, rather than reading $a$ and $n$ (or $k$ and $b$) off a graph as two separate un-logged numbers. The real mark scheme gives this its own two-step M1→dM1 dependency (a first correct log/index-law move, then the full elimination), and the examiner report names the dM1 as specifically where the question was most often lost even by otherwise-strong candidates: "the log rules not well understood by many students." One series is not enough to call this a recurring pattern, only a confirmed real question-type — stated at that strength and no further.

5.5 Interpreting a fitted constant in words, with no further calculation (§3.6). Confirmed in exactly 1 of the 9 series — Jun 2026 Q5(b), worth 1 mark on its own, no examiner report available yet (checked). The real mark scheme requires three components stated together (a direction word, the percentage to 2 s.f., and a per-year time unit) and names six near-miss wordings as explicitly not credited, each dropping exactly one of the three — e.g. a wording with the number and the unit but no direction ("it is changing by 19% per year"), or direction and unit but a wrong number, conflating the new total with the increase itself ("the annual increase is 119%," which is $b\times100$, not $(b-1)\times100$). The full six-item reject list is quoted verbatim in the existing lesson's own trap-taxonomy entry and is not reproduced again here. Because this series' constant happens to be $b\approx1.19>1$ (growth), the archive contains no real, verified instance of the decaying $b<1$ direction being examined this way — §4(f) above supplies an illustrative micro-example for that direction precisely because the real record doesn't yet contain one; this gap is restated in §6.

5.6 Base confusion / base-choice traps on a log-linear graph (§3.6). Confirmed in exactly 1 of the 9 series — Jun 2023 Q2, a log-log graph deliberately set in base 6 rather than base 10 or base $e$. The examiner report names the error directly: candidates "confused by log base 6" reached for base 10 or natural logs instead. The theory in §3.6 (gradient of a power-law fit is base-invariant; the intercept, and a semi-log fit's own gradient, are not) is exactly what this error breaks. A second, separate notation failure on the same real question — dropping the log notation entirely and writing a plain linear equation as if the axes weren't logged at all, or misplacing the base as a superscript — is also confirmed once, on the same question, not as an independently repeated pattern elsewhere.

5.7 Presentation traps that ride along with nearly every angle above, not specific to any one of them. Two are extremely well-evidenced across the whole WMA13 exponentials/logs archive (both already fully documented in the existing lesson's trap taxonomy, restated here only to place them correctly inside this synthesis rather than to re-argue them): a numerically correct modelling answer missing its unit or a contextual direction word, confirmed near-identically across at least 4 of the 9 series (Oct 2021, Jan 2022, Oct 2022, Jan 2025); and a correct final value reached by a calculator's built-in equation solver with no algebraic working shown, confirmed across at least 4 of the 9 series (Oct 2022, Jun 2022, Jan 2024, Jan 2025) with an explicit forward-looking warning in the Jun 2022 report that this may be penalised more strictly in future. A third, narrower one: the general "exact answer required, decimal given instead" rule lands specifically on this sub-topic whenever an equation of the form $e^{ax+b}=p$ resolves to a $\ln(\cdot)$-valued exact answer (general marking guidance, applying here rather than being unique to it).

5.8 A cross-topic cameo worth naming once. This sub-topic's own core solving mechanic — isolate the exponential term, take logs, using the stated domain constraint if one exists — recurs verbatim inside a real spec-1.2 (functions/composition) question, Jun 2026 Q2(d): $fg(a^2-5)=g(59)$ with $f(x)=3+4e^{2x}$, which collapses to exactly $3+4e^{2a}=8$, an instance of $e^{ax+b}=p$ solved for an exact answer in the form $\ln p$. This is filed under spec 1.2 in WMA13-verified-facts.md and codex-formula-and-method-bank.md (the composition/domain-constraint skill is what's actually being tested), and is not pulled into this document's own worked material for that reason — but it is worth recording here that the equation-solving half of that question is drawn from precisely the mechanism in §3.4, confirming the technique's reach extends past its own named spec item.

#6. What the record does not show — stated honestly rather than filled in

• No standalone "sketch the graph of eˣ / ln x / $e^{ax+b}+c$" mark-scheme item was located anywhere in the reviewed archive. Every real appearance of these graphs' shape is embedded inside a modelling or log-linear-fitting question (where the floor/ceiling or the axis-logging matters for reasoning, not a literal sketch mark), unlike spec 1.3's modulus graphs, which are directly, repeatedly tested as sketch-and-solve questions. Absence from a 9-series archive is evidence of rarity, not proof such a question has never appeared on this paper.
• The domain-restriction reasoning (§5.3) and the algebraic-elimination skill (§5.4) are each confirmed on exactly one real series. Neither should be described as "a recurring pattern" or given any implied frequency beyond "seen once in the 9 series checked" — doing otherwise would be inventing a likelihood this record doesn't support.
• The interpret-in-words skill (§5.5) has no verified real example of the $b<1$ decaying direction. §4(f)'s micro-example fills the theoretical gap but is explicitly not a past-paper fact.
• Jun 2026 has no examiner report yet. Every claim about that series in this document and in the existing lesson is sourced to the mark scheme and question paper only, checked as genuinely unavailable (404) rather than simply not looked for.
• Some real contexts named in §5.1 (Oct 2021 Q3, Oct 2021 Q7, Jan 2024 Q3) are recorded in the verified-facts archive only as located contexts, without a transcribed mark scheme or examiner-report quote pinning down exactly which marks are 3.1–3.3 versus 4.4. They are cited here as confirmed real contexts, not as sources for any specific mark-scheme mechanism, and no such mechanism is asserted for them above.

#7. What's additive here versus the rest of the WMA13 corpus

exponentials-and-logarithmic-graphs.ts already carries the full pedagogical treatment — prequestions, both mechanism derivations, two diagrams, three full worked chains (one VERIDIAN-original, two real past-paper reproductions), a chain-drill, a marked solution with a commonWrongPath, all seven traps, a beyond-spec block on why $e$ specifically, six MCQs and 21 flashcards — and is not repeated here. codex-formula-and-method-bank.md and forge-exam-technique-guide.md already carry a compressed formula/method skeleton for this sub-topic sized to sit alongside ten other WMA13 topics in one document. What this document adds: the seven-sub-case theory written out as one continuous, exam-anchored argument rather than pedagogy-paced blocks (§3); a named general diagnostic — "is the unknown a power of $x$, or is $x$ itself the exponent?" — for which axes to log, stated once as its own idea rather than left as two memorised recipes; the explicit three-way base-dependence rule (gradient of a power fit is base-invariant, gradient of an exponential fit is not, no intercept ever is) named as a single piece of theory rather than embedded inside a trap description; a frequency-scoped synthesis separating the ubiquitous 3.2 equation-solving skill from the comparatively rare 3.3 standalone log-linear question (§5.1–5.2), which no existing document states as an explicit contrast; and an honest inventory of exactly where the real record thins out (§6), including the one confirmed-absent question type (a bare sketch mark) and the one theoretically-necessary but not-yet-real-example sub-case (§4f/§5.5).