Topic Brief: Exponentials and Logarithmic Graphs

Provenance: compiled 2026-09-29, grepped and read directly against research/veridian/WMA13-verified-facts.md §5.3 ("Exponentials and logarithms," the primary, provenance-checked source for this topic) and §9's dated lesson-audit entry for this lesson, cross-checked against this folder's signal-frequency-and-boundaries.md and codex-deep-dive-exponentials-and-logarithmic-graphs.md so this brief does not restate their content, and against content/veridian/question-bank/wma13.ts (grepped — confirmed to hold no spec-3.1–3.3 item; its only exponential entry is a spec-4.4 differentiation question, Oct 2024 Q8(c), out of this brief's scope). No new primary-source PDF was independently pulled for this pass — this is a synthesis of what the verified-facts file already establishes, reorganised as a single fast "what to expect" reference for this one spec point.

#SIGNAL Topic Master Brief: Exponentials and Logarithmic Graphs (WMA13, spec 3.1–3.3)

#What this document is

The lesson (exponentials-and-logarithmic-graphs.ts) teaches the seven sub-case mechanics and worked chains. The CODEX deep-dive builds the full theory and a rich per-angle synthesis. This brief does neither — it is the short version: which real series tested this topic, at what question and tariff, in what recurring shape, and what to expect if it appears again. Full theory and workings live in the CODEX deep-dive; this brief only cites and classifies.

#Frequency

Reusing signal-frequency-and-boundaries.md's own topic-frequency table rather than recomputing it: real-world exponential/log modelling (spec 3.1–3.2) is seen in 8 of the 13 WMA13 series checked — Oct 2021, Jan 2022, Jun 2022, Oct 2022, Jun 2023, Jan 2024, Jan 2025, Jun 2026. Standalone log-linear-graph parameter estimation (spec 3.3) is seen in 2 of the 13 — Jun 2023 and Jun 2026, both of which also carry a 3.1–3.2 modelling question elsewhere in the same paper. Every one of these eight series was directly re-checked against WMA13-verified-facts.md §5.3 for this pass, not carried over unverified.

#The anchors, cited by series and question

#Mark-tariff pattern

Where the record cleanly isolates marks for this sub-topic, tariffs are small and specific: the one confirmed domain-restriction question is a single B1 (Jan 2024 Q5(d)); the two confirmed standalone log-linear questions are 6 marks (Jun 2023 Q2, all in scope) and 5 marks (Jun 2026 Q5(a)–(b) — a further 2 marks in Q5(c) belong to spec 4.4 and aren't counted here).

For the eight modelling questions, no clean isolation is possible: WMA13-verified-facts.md does not break several of them (Oct 2021 Q3/Q7, Jan 2024 Q3, Jan 2025 Q5) into a 3.1–3.2 share versus a 4.4 share, since one multi-part question routinely mixes equation-solving with a rate-of-change or long-run-behaviour part. signal-frequency-and-boundaries.md calls this combined question type "often the paper's largest single question, 9–12 marks," reused here rather than re-derived.

#Recurring context and angle type

Every 3.1–3.2 anchor is a real-world growth-or-decay story — mining, tree biology, medicine, pond ecology, temperature, finance, household waste — never bare algebra as a standalone question. The move is always the same: isolate the exponential or log term, apply the inverse function once, solve the linear expression it collapses to.

Spec 3.3 is at least three distinct sub-tasks, not one skill repeated: (i) reading a gradient and intercept off a log-linear graph as two separate un-logged numbers (Jun 2023 Q2(a), Jun 2026 Q5(a)); (ii) eliminating logs from an already-stated linear log-equation algebraically in one derivation, with its own M1→dM1 dependency (Jun 2023 Q2(b) only — the examiner report names this step as where marks are lost even by strong candidates); (iii) interpreting an already-found constant in words with zero further calculation (Jun 2026 Q5(b) only, needing a direction word, a percentage to 2 s.f., and a per-year unit together for the one mark).

#If this topic appears — what to expect

Expect a real-world growth-or-decay wrapper, not bare manipulation, possibly mixed with a spec-4.4 differentiation part in the same question. Show the exponential/log algebra explicitly — a calculator-only solve has cost marks despite a correct answer, confirmed across several series. If a log-linear graph appears, check which axis is logged, watch for a base other than 10 or $e$, and expect any of the three 3.3 sub-tasks above, not just "read two numbers off the graph." State the correct unit and a direction word on every final modelling answer — the most consistently repeated error in this archive. If asked to interpret a constant verbally, give direction, percentage, and time unit together; dropping any one has cost the mark in the one real instance on record.

#Honest gaps

No standalone "sketch $e^x$ / $\ln x$ / $e^{ax+b}+c$" mark-scheme item was located anywhere in the reviewed archive — every real appearance of these graphs' shape sits inside a modelling or log-linear question. The domain-restriction reasoning (Jan 2024 Q5(d)) and the log-elimination sub-task (Jun 2023 Q2(b)) are each confirmed on exactly one series, not described as recurring patterns beyond that. The interpret-in-words sub-task (Jun 2026 Q5(b)) is likewise confirmed once, and only for a growing ($b>1$) constant — no verified decaying ($b<1$) instance exists in the archive. Jun 2026 has no examiner report yet; every claim about that series here is sourced to its mark scheme and question paper only. No likelihood, percentage, or grade-boundary claim is made above.

#Sources

• research/veridian/WMA13-verified-facts.md — §5.3 for every anchor's question text, mark scheme, and examiner-report quote; §1 for the spec's own 3.1–3.3 wording; §9 for the dated lesson-audit entry covering the Jun 2026 gap.
• signal-frequency-and-boundaries.md (this folder) — the "8 of 13" and "2 of 13" frequency counts, reused rather than recomputed.
• codex-deep-dive-exponentials-and-logarithmic-graphs.md (this folder) — the full seven-sub-case theory, base-dependence rule, and mark-by-mark synthesis; consult it for workings, not this brief.
• content/veridian/curriculum/wma13/exponentials-and-logarithmic-graphs.ts — the taught mechanisms and worked chains, not reproduced here.
• content/veridian/question-bank/wma13.ts — checked; holds no spec-3.1–3.3 item.