TMUA Exponentials and Logarithms: What It Covers, How Much It Weighs, How to Fix It

Exponentials and logarithms is the TMUA mathematics topic coded MM5, covering the laws of logarithms, solving exponential equations of the form a^x=b, and the graph of y=a^x, three syllabus points in all, with the official syllabus explicitly placing the change of base formula outside what is tested. As catalogued by FrontierVUE, 102 TMUA questions are tagged to this topic; 25 of them come from past papers, with every year from 2016 to 2023 carrying some. The difficulty sits mainly at 47 medium and 39 easy, with only 11 rated hard and nothing at all in the very hard band. In other words this is a baseline that turns up in every catalogued year but rarely one that separates candidates: it decides not how high you can reach but whether you drop marks where you were meant to score. This page sets out the syllabus scope, the volume and difficulty shape, and a practice route you can follow as written.

TMUA Exponentials and Logarithms at a Glance

Topic code: MM5, a standalone topic in the TMUA mathematics syllabus, made up of 3 syllabus points
What it tests: The laws of logarithms, solving equations of the form a^x=b, and the graph of y=a^x
Explicitly out of scope: The change of base formula sits outside this topic's tested scope
Volume (FrontierVUE catalogue): 102 questions; 25 from past papers, every year from 2016 to 2023 carrying some
Difficulty shape (FrontierVUE catalogue): 47 medium, 39 easy, 11 hard and 5 very easy; the very hard band does not appear
How to treat it: A must-secure baseline rather than a differentiator: the target is fast and error-free

The scope and the split into syllabus points come from the mathematics syllabus published on the UAT-UK official TMUA test pages. The question counts and difficulty ratings are annotations in the FrontierVUE catalogue, not official paper statistics, since no official per-question topic labels are published.

What TMUA exponentials and logarithms actually covers

In the TMUA mathematics syllabus, exponentials and logarithms is a standalone topic coded MM5, built from three syllabus points: the laws of logarithms, solving exponential equations of the form a^x=b, and the graph of y=a^x. They are not three parallel blocks but a chain. You recognise the shape of an exponential function first, then master the laws that let you move between exponential and logarithmic form, then use those laws to solve equations. The syllabus also draws an unusually explicit boundary, placing the change of base formula outside what this topic tests, so converting between bases is not something you need to prepare for.

MM5 Exponentials and logarithms

The 3 syllabus points the TMUA specification lists under this topic

MM5.2 Laws of logarithmsThe backbone of the topic

Moving between exponential and logarithmic form
Swapping sums and differences of logs for products and quotients
Moving a coefficient into an exponent and back out again
Special cases such as the log of a reciprocal and the log of the base itself
Change of base is not tested

MM5.3 Solving exponential equationsWhere those laws actually get used

Solving equations already in the form a^x=b
Rearranging first, then reducing to that form
Exponential equations that become quadratics after a substitution

MM5.1 The graph of an exponentialThe foundation the other two rest on

The shape and behaviour of y=a^x for simple positive bases
Telling growth apart from decay

Tap a branch to unfold

The change of base formula is explicitly outside what this topic tests. That boundary is useful: it means you do not need to prepare for converting between different bases, and the whole weight of revision belongs on the same-base laws instead, until swapping between sums and products, or between coefficients and exponents, needs no thought at all. The difficulty here is usually not remembering the handful of laws, it is carrying them through a long chain of rearrangements without a single slip.

How much weight this topic carries in TMUA

Two things tell you whether a topic deserves your time: how often it turns up, and how much of the question bank it occupies. As catalogued by FrontierVUE, 102 TMUA questions are tagged to exponentials and logarithms, and 43 of those appearances sit on mock papers. One caveat matters: these are FrontierVUE catalogue annotations rather than official paper statistics, because the official specification lists syllabus content rather than tagging individual questions.

102

Questions on this topic in the FrontierVUE catalogue

Of those, appearances on mock papers

Syllabus points listed under the topic

2016 to 2023

Every past paper year in that span carries some

	Questions in the FrontierVUE catalogue	Its role within the topic
MM5.2 Log laws	64	The backbone: the largest of the three counts
MM5.3 Exponential equations	44	The main application: where the laws get cashed out
MM5.1 Exponential graphs	16	The foundation: the lightest count, but the other two are built on it

A single question can be tagged to more than one syllabus point, so these counts are not meant to line up with the topic total. All figures are annotations in the FrontierVUE catalogue rather than official paper statistics.

The spread across years is worth a look too. Among the past papers catalogued by FrontierVUE, the topic shows up in every year from 2016 to 2023 with no year missing, the smallest count falling in 2018. A further 6 questions sit on the official specimen papers, which belong to no sitting year; the first live TMUA sitting was in 2016. Turning up in every one of those years marks it as a settled fixture rather than the preference of one or two sittings.

Year	Questions in the FrontierVUE catalogue
2016	3
2017	4
2018	2
2019	3
2020	3
2021	3
2022	3
2023	4
Official specimen papers (no sitting year)	6

Where it gets hard, and why losing marks here is a bad trade

The shape of a difficulty distribution usually tells you more than an average does. As catalogued by FrontierVUE, 47 questions on this topic are rated medium, 39 easy, 11 hard and 5 very easy, with nothing at all in the very hard band. The centre of that curve sits firmly on the easier side, and the implication is blunt: exponentials and logarithms is not where candidates get separated, it is the baseline everyone is assumed to clear.

47 medium: the mainstream shape here

Medium is the largest single band here in the FrontierVUE catalogue. These questions rarely hide the idea; the difficulty is the number of steps. You bring everything onto a common base, apply the laws two or three times in a row, and only then land on something you can actually solve. Each step looks easy on its own, and the chain is where it breaks.

39 easy and 5 very easy: the band you can least afford to drop

Both of these bands are low-difficulty questions (39 easy and 5 very easy in the FrontierVUE catalogue), and they are the ones you can least afford to get wrong. Low difficulty means a wrong answer has no excuse and no way back. TMUA is multiple choice, so a slip and a blank look identical on the paper. Driving these two bands to zero errors is worth far more than grinding through a few extra hard ones.

11 hard, and nothing very hard at all

This is the feature worth remembering (FrontierVUE catalogue): only 11 questions are rated hard, and the very hard band is empty. So do not sink hours here hunting for difficulty. The ceiling on exponentials and logarithms is low by nature, and effort spent above that ceiling buys you nothing extra.

Three slips we run into often when teaching this

First, the domain. The argument of a logarithm must be positive, so after rearranging you have to go back and check that each solution is still admissible; when several values come out, one of them often has to be discarded. Second, applying a law backwards: treating the log of a sum as the sum of logs. Third, forgetting to undo a substitution: once a substitution turns the equation into a quadratic and you solve it, what you are holding is the intermediate quantity, not the original unknown.

Set the goal here as fast and error-free, not as conquering hard problems. The difficulty distribution has already drawn the ceiling for you. What decides your score is whether you can take these questions without burning time and without slipping, so that the minutes you save go to the topics that genuinely separate candidates.

How to close the gap on this topic

The order for fixing this topic should follow dependency rather than syllabus numbering: get the graphs solid first, drill the laws until they are automatic, and only then work on reducing equations. The route below can be followed as written, and every step comes with a finish condition; if you cannot meet it, do not rush on to the next one.

01MM5.1

Set the graph foundation first

Check that you can sketch how y=a^x behaves for a simple positive base, tell growth apart from decay, and say how the curve sits relative to the axes. Finish condition: you draw it correctly with nothing to look at.

02MM5.2

Drill the laws until they are automatic

Work through same-base conversions between sums and products, moving coefficients into exponents and back, and special cases such as the log of a reciprocal or the log of the base itself. Finish condition: given a string of logarithms you write the tidied result straight out, without pausing.

03MM5.3

Practise recognising equation shapes

The point is not the solving as such but the recognition: when to take logarithms of both sides, when a substitution turns the thing into a quadratic, and when getting everything onto a common base is already enough. Finish condition: you name the shape before you start writing.

04Timed

Timed sets, aimed at low-difficulty slips

In our teaching experience, slips on low-difficulty questions tend to surface only under time pressure. Use topic practice to pull questions on this topic, work them in timed sets, and log every miss under one of three headings: a domain check, a law applied backwards, or a substitution never reversed.

05Full paper

Put it back into a full paper

Being fluent in isolation is not the same as being steady in an exam. Go back into complete mock papers and see whether questions on this topic start failing again once earlier questions have eaten your clock. As catalogued by FrontierVUE, this topic appears 43 times on mock papers, which is enough material for this step.

This page is open to read. Practice and the weakness report require you to sign in, and the weakness report also needs the relevant subject unlocked. Actually working the topic means pulling questions by topic in the practice centre.

FAQ

Frequently asked questions

Is TMUA exponentials and logarithms hard?

Not hard overall, but expensive to get wrong. As catalogued by FrontierVUE, 47 questions on this topic are rated medium, 39 easy, 11 hard and 5 very easy, with nothing in the very hard band. A centre of gravity at medium and below marks it out as a baseline everyone is expected to clear. In our teaching experience marks tend to go missing not because a question is hard but because the domain of a logarithm went unchecked, a law was applied backwards, or a substitution was never reversed.

Does TMUA test the change of base formula?

No. Under exponentials and logarithms, the TMUA mathematics syllabus states that change of base is not tested. In practice that means you do not need to prepare for converting between bases, but the same-base laws have to be second nature: swapping sums and differences for products and quotients, moving coefficients into exponents and back, and special cases such as the log of a reciprocal or the log of the base itself.

Does exponentials and logarithms come up in TMUA every year?

Among the past papers catalogued by FrontierVUE, the topic appears in every year from 2016 to 2023: 3 questions in 2016, 4 in 2017, 2 in 2018, 3 in 2019, 3 in 2020, 3 in 2021, 3 in 2022 and 4 in 2023, with a further 6 on the official specimen papers, which belong to no sitting year; the first live TMUA sitting was in 2016. These are FrontierVUE catalogue annotations, not official paper statistics, since no official per-question topic labels are published, so they show the distribution across what is catalogued rather than an exact count on any year's paper.

How should I practise TMUA exponentials and logarithms?

Follow dependency, not syllabus numbering. First make sure you can draw y=a^x correctly with nothing to look at, then drill the same-base log laws until a tidied result comes out at a glance, then practise recognising the shape of an exponential equation: when to take logarithms of both sides, when a substitution turns it into a quadratic. After those three, work timed sets and file every mistake under a domain check, a law applied backwards, or a substitution never reversed, then put the topic back into a complete mock paper.

Sources

Official sources

By topic

Practise this TMUA topic

Filter the question bank down to this topic and work through it in a timed interface with worked solutions. Sign up free to start; the first 2 papers of each subject are free.

Start practising All TMUA papers