Deep Dive: Logarithms and Exponentials

Provenance (2026-09-29): read in full — research/veridian/WMA12-verified-facts.md (§1–§9, grepped and re-read directly for every logarithms/exponentials citation: §2 spec transcription, §3 formula-booklet contents, §4 mark-scheme conventions, §6.6, §7, §8, and the logarithms-and-exponentials.ts entries in the §9 lesson-audit log, including the 2026-09-14 follow-up pass on part (c)); content/veridian/curriculum/wma12/logarithms-and-exponentials.ts (the real lesson file, read in full, specifically to identify what it already teaches and avoid duplicating its worked chains, chain drill, method comparison, marked solution, or its own six trap-taxonomy items); content/veridian/technique-hub/wma12/codex-formula-and-method-bank.md, forge-exam-technique-guide.md, and signal-frequency-and-boundaries.md (all read in full, to avoid restating their tables); content/veridian/question-bank/wma12.ts (grepped — contains no logarithms/exponentials question yet). No new primary-source PDF was opened for this pass; every fact below traces to one of the documents just named, which themselves trace to the primary Pearson PDFs. All arithmetic in every micro-example below was independently computed and cross-checked with Python before being written in.

#CODEX Deep-Dive: Logarithms and Exponentials (WMA12, spec 5.1–5.3)

#0. What this document is, and what it deliberately doesn't repeat

Three documents already cover this topic from three other angles: the lesson file (logarithms-and-exponentials.ts) teaches it end-to-end with full worked chains and six named traps; the CODEX formula bank gives the formula table and two short method skeletons; SIGNAL gives the raw frequency count (seen in 5 of the 20 series with a question paper and mark scheme individually reviewed so far) and the mark range (3–10). None of the three lays out, in one place, every distinct real question-shape this spec area has actually been tested through, with the theory each shape depends on and a short from-scratch example of each. That gap is this document's job. Where a fact is already fully taught elsewhere, this document points at it rather than re-teaching it.

#1. Where this sits on the paper

Spec item 5, "Exponentials and logarithms," has three sub-points — 5.1 (the graph of $y=a^x$), 5.2 (five laws of logarithms), 5.3 (solving $a^x=b$, "students may use the change of base formula") — confirmed against the specification's own numbered content list (verified-facts §2). Of everything spec 5 requires, only the change-of-base formula is printed on the P2 formula-booklet page; the graph shape and all five laws have to be recalled from memory (verified-facts §3). A precise nuance the CODEX formula bank found by reading the spec's own "Notation and formulae" section directly: that section's explicit "must-know, not in the booklet" callout for P2 re-lists only three of the five laws (addition, subtraction, power) — the reciprocal and identity laws are still spec 5.2 content, just not separately re-flagged there, because both fall out of the other three in one line (codex-formula-and-method-bank.md, §5).

The topic has been individually confirmed tested in 5 of the 20 series with a full question paper and mark scheme reviewed to date, at a real observed mark weight of 3–10 marks (signal-frequency-and-boundaries.md, Part 1) — a floor on what has been specifically cited, not a rarity signal, since every spec 5 sub-point is compulsory content.

#2. The complete theory

#2.1 The exponential function, $y=a^x$

Spec 5.1 restricts $a$ to $a>0,\ a\neq1$. Both restrictions have a concrete reason, not just a domain footnote. If $a<0$, $a^x$ has gaps in its domain — $(-4)^{0.5}$ has no real value — so the function would not be defined for every real $x$. If $a=1$, $1^x=1$ identically: a horizontal line, not a growth/decay curve. Given a valid $a$: $a^0=1$ always, so every curve in the family passes through $(0,1)$; $a^x>0$ for every real $x$ (a product of positive factors is positive), so the curve has a horizontal asymptote at $y=0$ and, critically, no $x$-intercept at all — $a^x=0$ has no solution, ever. The curve is strictly increasing when $a>1$ and strictly decreasing when $0<a<1$, with no third case, because $a>0,\ a\neq1$ partitions every valid base into exactly those two behaviours.

Because the function is strictly monotonic across its whole domain, it never repeats an output — which is exactly what makes "what power of $a$ gives this number?" a question with at most one answer for any positive number, which is the entire justification for defining a logarithm at all (§2.2).

Scope boundary, confirmed by absence, not inferred: nothing in the specification's P2 content, the P2 formula-booklet page, or any of the 20 series' worth of examiner reports and mark schemes reviewed for the facts bank mentions $e$ or $\ln$ as a distinguished base for this unit (a direct re-grep of the facts bank for "natural log," "ln," or "base e" for this pass returns zero matches). $a$ in spec 5.1 is a general positive base — a calculator's $\log$ or $\ln$ button is a computational tool for evaluating any base via change-of-base (§2.4), not evidence that $e$ itself is examinable content at P2. A content writer or student bringing $e^x$/$\ln x$ intuition from a different course's exponential-growth unit should not expect WMA12 to test $e$ specifically.

#2.2 The logarithm, defined as the inverse of §2.1

$\log_a b = c$ means, by definition and nothing more, that $a^c=b$ (for $a>0,\ a\neq1,\ b>0$). The condition $b>0$ is not a separate rule to memorise — it falls straight out of §2.1: $a^c$ is always positive, so there is no power of $a$ that produces zero or a negative number, and asking for $\log_a$ of one is a question with no answer, not a hard question. Every law in §2.3 and the whole method for solving $a^x=b$ in §3 is nothing more than this one definition applied to a specific equation.

#2.3 The five laws, derived from the laws of indices

Let $\log_a x=p$ and $\log_a y=q$, i.e. $a^p=x$ and $a^q=y$. Multiplying the two equations and using $a^p\times a^q=a^{p+q}$ (indices add under multiplication) gives $xy=a^{p+q}$, so $\log_a(xy)=p+q=\log_a x+\log_a y$ — the addition law. Dividing instead gives $x/y=a^{p-q}$, so $\log_a(x/y)=\log_a x-\log_a y$ — the subtraction law. Raising the first equation to the power $k$ gives $x^k=a^{pk}$, so $\log_a(x^k)=k\log_a x$ — the power law, and the exact reason $(\log_a y)^2$ (square the number $\log_a y$) and $\log_a(y^2)$ (apply $\log_a$ to the squared argument) are structurally different expressions, not two notations for the same thing. The remaining two laws are special cases: $\log_a(1/x)=\log_a1-\log_a x=-\log_a x$ (since $\log_a1=0$, because $a^0=1$ for any valid $a$) — the reciprocal law; and $\log_a a=1$ — the identity law, immediate from the definition since $a^1=a$. None of the five is an independent fact; all five are the laws of indices, read through one definition.

#2.4 The change-of-base formula, derived

Given only a calculator that computes logs in one fixed base $b$ (base 10 or base $e$), and wanting $\log_a x$: let $c=\log_a x$, so $a^c=x$. Take $\log_b$ of both sides: $\log_b(a^c)=\log_b x$. The left side is the power law again, in base $b$: $c\log_b a=\log_b x$, so $c=\dfrac{\log_b x}{\log_b a}$ — i.e. $\log_a x=\dfrac{\log_b x}{\log_b a}$. It is not a sixth, independent fact — it is the power law applied through the definition a second time, which is exactly why it is the one identity Pearson prints (spec 5.3 names it by name: "students may use the change of base formula") and the other five are not: it is the identity that converts "evaluate a log in a base your calculator doesn't have a button for" into a calculation a $\log$/$\ln$ button actually performs. Its real, spec-confirmed use on this paper is narrower than "evaluate any log": it is named specifically under 5.3, solving $a^x=b$, where any consistent base — not necessarily one requiring the formula's own two-log-division form — is accepted (verified-facts §6.6's Jan 2020 "oe" example: "States or uses $\log_2 4x^2=2\log_2 2x$ (oe)").

#3. Every real distinct sub-case verified in the archive, with a short micro-example each

Six genuinely different question-shapes for spec 5 have individually named, real citations in the facts bank. Four are equation-solving shapes that differ in how many steps of §2.3/2.4 they chain together; two are cross-topic pairings where a log or an exponential is embedded inside a different topic's own technique rather than being the whole question. Every micro-example below is VERIDIAN-original (not a reproduction of any real question) and independently checked by substitution.

#3.1 Solve $a^x=b$ in clean index form, decimal answer

The simplest real shape: the exponential term is already isolated, nothing multiplies or divides it, and the question asks for a decimal to a stated precision. Verified anchor: Jan 2021 Q3(i), $7^{x+2}=3$ (verified-facts §6.6, lesson-audit log). Micro-example: solve $3^x=40$ to 3 s.f. Take $\log$ of both sides and apply the power law: $x\log3=\log40$, so $x=\log_340=3.36$ (3 s.f.) — checked: $3^{3.3578}\approx40.0$.

#3.2 Solve $a^x=b$, answer required as a single combined logarithm — both directions

A distinct, harder shape: the question specifies the answer form (e.g. "$x=\log_a k$, $k$ rational"), so a correct but unsimplified exact expression earns only an intermediate mark, and a further, separate mark is earned only by combining it. Confirmed exactly on Jan 2021 Q3(i): reaching $x=\log_73-2$ earned the first accuracy mark in full, but "did not reach the required form, failing to deal with the $-2$" cost the second, distinct one (verified-facts §6.6; the same question drives the lesson file's own single-log-form-left-uncombined trap). The combining step always works the same way — rewrite the leftover constant as a log via $\log_aa=1$, then apply the addition or subtraction law — but the real anchor happens to need subtraction (a constant is subtracted off the log term). The other direction, a constant added on, is equally real and uses the same technique the other way:

• Subtraction direction (matches the real anchor's shape). Solve $2^{x+3}=50$, answer as $x=\log_2k$. $x+3=\log_250$, so $x=\log_250-3=\log_250-\log_28=\log_2\frac{25}{4}$ (using $\log_28=3$ since $2^3=8$) — checked: $\log_250-3=2.6439=\log_2(25/4)$.
• Addition direction (the same skill, run the other way). Solve $5^{x-1}=8$, answer as $x=\log_5k$. $x-1=\log_58$, so $x=\log_58+1=\log_58+\log_55=\log_540$ — checked: $\log_58+1=2.2920=\log_540$.

#3.3 Combine two log terms with no leftover constant, reach a log-free equation, solve, reject an invalid root

The most heavily-anchored shape in the archive: four independent series test the identical structure — combine, remove the log, solve, reject — with forming the log-free equation confirmed as its own standalone accuracy mark, distinct from both the combining step before it and the solving step after it, on every one of Jan 2021 Q3(ii), Jun 2023 Q5, Jun 2024 Q3(i), and Jun 2025 Q6(i) (verified-facts' lesson-audit-log follow-up, 2026-09-14; corroborated in codex-formula-and-method-bank.md, §6). Micro-example: solve $\log_2(x+1)+\log_2(x-1)=3$. Combine (addition law): $\log_2[(x+1)(x-1)]=3$, so $x^2-1=8$ (its own standalone mark), giving $x^2=9$, so $x=\pm3$. The domain of the original equation needs $x-1>0$, i.e. $x>1$; $x=-3$ is rejected, $x=3$ accepted — checked: $\log_2 4+\log_2 2=2+1=3$.

#3.4 Combine three-plus log terms including a bare constant that must first become a log

The archive's single most demanding real logs question, and structurally distinct from 3.3: Jan 2021 Q3(ii) itself, $1+\log_2y+\log_2(y+4)=\log_2(5-y)$ — three log terms plus a leading constant, logs on both sides, an M1→dM1→A1→ddM1→A1 chain (5 of the question's 8 marks), where the mark scheme records the quadratic $2y^2+9y-5=0$ and states the root $y=-5$ "must have been rejected if seen," accepting only $y=\tfrac12$ (verified-facts §6.6 and lesson-audit log). The extra technique 3.3 doesn't need: the bare constant ($1$, here) has to be rewritten as $\log_2 2$ before the addition law can pull every term into one log. Micro-example, same shape, different numbers: solve $1+\log_2y+\log_2(y+3)=\log_2(4-y)$. Rewrite the constant: $\log_22+\log_2y+\log_2(y+3)=\log_2(4-y)$, combine: $\log_2[2y(y+3)]=\log_2(4-y)$, remove the log: $2y(y+3)=4-y$, i.e. $2y^2+7y-4=0$, giving $y=\tfrac12$ or $y=-4$. The original equation needs $y>0$ and $y+3>0$ and $4-y>0$ simultaneously; $y=-4$ fails every one of those and is rejected, $y=\tfrac12$ passes all three — checked: $1+\log_2(0.5)+\log_2(3.5)=\log_2(3.5)$ on both sides, $\approx1.8074$.

#3.5 Logs embedded inside a different topic's own technique

Two confirmed real pairings put a logarithm inside another spec area's machinery rather than making the whole question about logs:

• Logs plus sigma/summation notation. Jan 2023 Q4(ii) is described in the facts bank as "a harder sigma-notation-plus-logs question," verified as "one of the most challenging parts on the paper," with one specific confirmed error carried over from it: confusing $\log_a(y^2)$ with $(\log_ay)^2$ (verified-facts §6.6). The facts bank's own citation does not preserve the question's exact equation beyond that one error, so this document does not claim more detail than that — only that the pairing itself, and that one error, are real and verified. What the pairing requires, mechanically, whatever its exact real form was, is repeated use of the addition law across a sum of several log terms — worth illustrating directly: $\displaystyle\sum_{r=1}^{3}\log_2(2r)=\log_22+\log_24+\log_26=\log_2(2\times4\times6)=\log_2 48$ — checked: $1+2+2.58496=5.58496=\log_248$.
• Logs inside the trapezium rule. Jan 2025 tested "a novel way of testing the trapezium rule via logarithms," confirmed as a genuinely new format, not previously seen in the archive (verified-facts §6.9). Here the log function is the integrand being numerically approximated, not an equation being solved — the trapezium-rule mechanic itself (already fully taught in the-trapezium-rule.ts and not reproduced here) is unchanged; only the source of the $y$-values differs. Illustrative fragment only: four strips of $y=\log_2x$ across $x=1$ to $5$ ($h=1$) give $y$-values $0,\ 1,\ 1.58496,\ 2,\ 2.32193$, and $\frac{h}{2}\big[(y_0+y_4)+2(y_1+y_2+y_3)\big]=\frac12\big[2.32193+2(4.58496)\big]\approx5.746$.

#3.6 A sub-case with no verified real instance — named as a gap, not filled

Per this course's standing no-fabrication rule: a direct re-grep of the entire facts bank for "sketch," "transformation," "growth," "decay," "half-life," or any modelling context attached to an exponential turns up zero matches. There is no verified real WMA12 question asking a student to sketch or transform $y=a^x$, or to build a growth/decay context around it — spec 5.1 tests the graph's properties (§2.1) as background for solving equations, not as a sketching exercise in its own right, in every instance found so far. This is reported as an honest absence in the archive reviewed to date (20 of an unknown total number of series, per §5's own caveat below), not a claim that such a question can never appear.

#4. The trap taxonomy, reorganised by where in the mark ladder it strikes

The lesson file already carries six fully-developed trap-taxonomy items, each with its own verbatim, attributed citation; this section adds no new trap, only a synthesis the lesson, CODEX, and SIGNAL don't currently give — which stage of a real mark scheme each one actually costs, since that determines how much of a question survives the error.

The practical reading: the six real traps this topic has produced are not evenly spread across a question — three of them cluster at the log-law-application step, and the two "form-matching"/"root-checking" failures both strike only at the very last mark, after every other mark has already been safely banked, which is exactly why real scripts show sharp mark-by-mark drop-offs late in these questions rather than early wholesale failure (verified-facts §6.6: "nearly 90% to just over 70%... only 50% scoring all three," Jan 2021 Q3(i)).

#5. Every real, individually-verified instance of this topic — one consolidated table

Collated from verified-facts §6.6, its lesson-audit-log follow-up, and the CODEX/SIGNAL cross-citations — the one place this course's research currently puts every named real instance of spec 5 in a single table, rather than split across three documents' prose. Note this table is wider than SIGNAL's own "5 of 20" frequency count: SIGNAL's row is scoped to citations already attached to the logarithms-and-exponentials.ts lesson file specifically (§6.6's own five series); the Jan 2025 row below is drawn from §6.9 (trapezium rule) instead, where a log function appears as the content of a different topic's question (§3.5) — a real, additional instance of spec 5 being examined, just not one SIGNAL's per-lesson-file count was scoped to catch.

Blank "Marks" cells above are not zero — they mean the facts bank's own citation for that finding quoted the examiner-report prose without separately re-extracting that question's per-line mark allocation in this pass; nothing here should be read as claiming those questions were worth zero marks.

#6. What remains open — reported, not smoothed over

• The "5 of 20" count is a floor, not a rate. Only 20 series (Jun 2019–Jan 2026) have a question paper and mark scheme individually reviewed for this facts bank at all, and no pass has topic-tagged all 75 marks of all 20 (verified-facts §5; SIGNAL Part 1's own methodology note). Further named instances of spec 5 almost certainly exist in the 15 series not yet cited by name here.
• October 2025 and January 2026 have no examiner report in any archive searched — very likely simply not yet published, not a confirmed permanent absence (verified-facts §8).
• §3.6's absence is a gap in what has been checked, not a proof of what Pearson can ask. No likelihood, percentage, or grade-boundary claim is made anywhere in this document beyond the real, already-verified examiner-report percentages quoted by name in §4 and §6.6 of the facts bank — this document adds no new predictive figure of its own, per this course's standing rule.
• Jan 2023 Q4(i)/(ii)'s own per-line mark allocation was not independently re-extracted in this pass (only the examiner-report prose was available from the facts bank's own §6.6 citation) — a future pass with the raw mark-scheme PDF could add the M/A/B structure the way the Jan 2021 and Jun 2023–2025 anchors above already have it.

#Sources

• research/veridian/WMA12-verified-facts.md — §1 (unit identity), §2 (spec transcription), §3 (formula-booklet contents), §4 (mark-scheme conventions), §5 (archive scope), §6.6 (Exponentials and logarithms), §7–§8 (topic-strength and flagged-gap summary), and the logarithms-and-exponentials.ts entries of the §9 lesson-audit log (original pass and the 2026-09-14 part-(c) follow-up).
• content/veridian/curriculum/wma12/logarithms-and-exponentials.ts — the real lesson file, read in full to confirm what it already teaches and avoid duplicating its worked chain, chain drill, method comparison, marked solution, or six trap-taxonomy items.
• content/veridian/technique-hub/wma12/codex-formula-and-method-bank.md — §5 (formula table and the spec's own "Notation and formulae" must-know distinction) and §6 (method skeletons), read in full.
• content/veridian/technique-hub/wma12/forge-exam-technique-guide.md and signal-frequency-and-boundaries.md — read in full for cross-checking; SIGNAL Part 1's frequency row and June-2025 per-question table are cited, not reproduced.
• content/veridian/question-bank/wma12.ts — grepped for existing logarithms/exponentials coverage; none found.