Topic Brief: Logarithms and Exponentials

Provenance (2026-09-29). Read in full for this pass: research/veridian/WMA12-verified-facts.md (§2's verbatim transcription of spec item 5; §3's formula-booklet section; §6.6 in full; §7's "topics with enough verified material" entry for this topic; §9's logarithms-and-exponentials.ts lesson-audit-log entry and its 2026-09-14 part-(c) follow-up pass, which independently re-extracted Jun 2023 Q5, Jun 2024 Q3(i) and Jun 2025 Q6(i) from the raw mark-scheme PDFs). Also read in full: content/veridian/curriculum/wma12/logarithms-and-exponentials.ts (the real lesson file, to confirm no citation exists beyond what the facts bank already gives); codex-deep-dive-logarithms-and- exponentials.md (all six question-shapes, its consolidated instance table, and its trap-taxonomy mark-ladder section — not repeated here); codex-formula-and-method-bank.md §5–6; forge-exam- technique-guide.md (grepped for "log"); signal-frequency-and-boundaries.md (this topic's own row and the June 2025 per-question table). content/veridian/question-bank/wma12.ts was grepped for "log"/"exponential" — zero matches; no worked past-paper item exists there for this topic yet. Not affiliated with or endorsed by Pearson Edexcel. Every quotation below is short (under ~15 words) and attributed to a named document; no full question, mark-scheme extract, or examiner-report passage is reproduced at length.

#SIGNAL Topic Master Brief — Logarithms and Exponentials (WMA12, spec 5.1–5.3)

#What this is, and isn't

A single-topic zoom-in on the frequency/expectation layer for one spec item, in the same narrow lane as the binomial-expansion SIGNAL brief. signal-frequency-and-boundaries.md gives this topic one row in a paper-wide table; codex-deep-dive-logarithms-and-exponentials.md proves the theory, works six distinct question-shapes from scratch, and stages the six real traps against the mark ladder; the lesson file teaches it end-to-end. This document repeats none of that. It pulls only the frequency/pattern/expectation facts already established, organises them as "what to expect before opening the question," and adds nothing invented.

#Real frequency: seen in 5 of 20 series checked

The archive has 20 series with a question paper and mark scheme read (June 2019–January 2026) and 17 with a published examiner report (WMA12-verified-facts.md §5). A dedicated logs/ exponentials question — one where spec 5 is the whole question, not a fragment inside another topic's own machinery — is individually confirmed in 5 of those 20 series: January 2021, January 2023, June 2023, June 2024, and June 2025. This matches the count already used in signal-frequency- and-boundaries.md and codex-deep-dive-logarithms-and-exponentials.md exactly; it is the same floor restated at topic scale, not a new count. A floor of 5 does not mean the topic is uncommon — spec 5 is compulsory, recurring content — it means only that five series have been individually verified by citation so far; the other 15 have not yet been topic-tagged in full.

A sixth series carries the topic too, but not as a standalone question: January 2025 embedded a logarithm as the function being numerically approximated inside a trapezium-rule question — confirmed by the facts bank as "a novel way of testing the trapezium rule via logarithms" (§6.9). This is a real, additional appearance of spec 5 content, just not one the "5 of 20" count above was scoped to catch, since that count is specifically dedicated logs questions.

#Every real series/question that has tested this topic

No further real citation for this topic exists in any document read for this pass.

#Real mark-tariff pattern

signal-frequency-and-boundaries.md records the topic's confirmed range as 3–10 marks. Where the individual question is a standalone equation-solve, the tariff runs from a flat 3 (Jan 2021 Q3(i), M1 A1 A1, no dependency chain) up to a 5-mark combine-and-solve chain with a genuine dependency structure (B1/M1/A1/dM1/A1 on Jun 2025 Q6(i); M1/dM1/A1/ddM1/A1 on the harder Jan 2021 Q3(ii)). Every question number confirmed above — Q3, Q3, Q4, Q4, Q5, Q3, Q6 — falls in the Q3–Q6 band; the topic has not been individually cited opening a paper (Q1–Q2) or closing one in any series read for this pass, unlike binomial expansion, which the facts bank records opening June 2025's paper outright.

#Real recurring context/angle across the series checked

Three genuine patterns recur across the citations, not one:

• "Reaching an answer" and "finishing it into the exact required form" are consistently two separate marks, in both directions this topic takes. For a solved $a^x=b$ equation, a correct but unsimplified exact expression earns an intermediate accuracy mark; only combining it into the specific form the question names earns a further, distinct one (Jan 2021 Q3(i)). For a combined-log equation, forming the log-free equation is confirmed as its own standalone accuracy mark, separate from both the combining step before it and the solving step after it, on four independent series (Jan 2021 Q3(ii); Jun 2023 Q5; Jun 2024 Q3(i); Jun 2025 Q6(i)). Both are instances of the same underlying pattern: real WMA12 mark schemes on this topic reward the intermediate step explicitly, not just the final number.
• Whenever logs are combined and solved as an equation, rejecting an invalid root is an explicit, named final-mark requirement, not an optional flourish. All four "combine, remove, solve" anchors above state this in the mark scheme's own final line — a root that makes an original log's argument invalid "must have been rejected if seen."
• This topic is regularly tested embedded inside a different topic's own technique, not only in isolation. Confirmed twice: Jan 2023 Q4(ii) paired sigma notation with logs, and Jan 2025 embedded a log function inside a trapezium-rule question, the latter explicitly flagged by the examiner report as a novel format for this paper.

#If this topic appears: what to expect

• Expect it in the Q3–Q6 range of the paper, worth 3–10 marks — not the opening question, and not necessarily a single self-contained item, since it has twice been folded into a different topic's own machinery.
• Expect one of two dominant real shapes: a straight "solve $a^x=b$" (isolate the exponential term, take logs, rearrange), or "combine several log terms into one, remove the log, solve the resulting equation, and reject any invalid root."
• If the question names the exact form of the final answer (e.g. "$x=\log_ak$, $k$ rational"), expect a real, separate accuracy mark for actually combining the expression into that form — a correct but unsimplified expression earns only the earlier mark.
• If two or more logs are being combined into an equation, expect "form the correct log-free equation" to be its own standalone mark, and expect the final mark to explicitly require rejecting any root that is invalid in the original equation.
• Watch for two specific, named recurring slips if logs are combined: treating log subtraction as division of the log terms themselves, and distributing a log across a sum inside its own argument.
• Do not assume this topic is tested in isolation — it has been paired with sigma notation once and with the trapezium rule once, in the series checked.

#What this document deliberately leaves out

No derivation of the five log laws or the change-of-base formula, and no worked mechanism for any of the six question-shapes (codex-deep-dive-logarithms-and-exponentials.md, §2–3). No full trap taxonomy staged against the mark ladder (same document, §4) and no general exam-timing or booklet- contents material (forge-exam-technique-guide.md). No percentage, likelihood, or grade-boundary claim of any kind — frequency is stated only as "seen in N of M series checked," per this course's standing discipline; paper-wide grade boundaries live in signal-frequency-and-boundaries.md Part 2, and none is topic-specific.

#Sources used

• research/veridian/WMA12-verified-facts.md — §2, §3, §5, §6.6, §6.9, §7, and the §9 logarithms-and-exponentials.ts audit entries (original pass and the 2026-09-14 part-(c) follow-up), read in full.
• content/veridian/curriculum/wma12/logarithms-and-exponentials.ts — read to confirm no citation exists beyond the facts bank's own.
• codex-deep-dive-logarithms-and-exponentials.md, codex-formula-and-method-bank.md (§5–6), forge-exam-technique-guide.md (grepped for "log"), signal-frequency-and-boundaries.md (this topic's row and the June 2025 table) — read to confirm nothing above restates them.
• content/veridian/question-bank/wma12.ts — grepped for "log" and "exponential"; zero matches.