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UAT-UK SPEC: MM5MODULE: MATHEMATICS 2TOPIC ID: ARCH-MM-07

Laws of Logarithms, Disguised Exponential Quadratics (a²ˣ + b aˣ + c = 0) & Log Inequalities

Master Laws of Logarithms, Disguised Exponential Quadratics (a²ˣ + b aˣ + c = 0) & Log Inequalities on the ESAT Mathematics 2 paper. Traditional classroom derivations waste valuable seconds. Using the 2-step Sub-90 invariant elimination method compresses solve time to 40s, saving +110s per question.

Standard Classroom Time
150s
Textbook multi-line derivations bleed exam clock
Sub-90 Target Pace
40s
Comfortably below the 88.8s per-question limit
Time Saved for Review
+110s
Banks clock buffer across 27 questions

2-Step Sub-90 Shortcut Formula

How to eliminate intermediate algebra and solve under 45 seconds.

Launch Interactive Lesson
STEP 1 • SETUP INVARIANT (LINE 1)10–15s
Line 1 (Telescoping Log Product): All interior numerators and denominators cancel, leaving log⁡2 ⁣(last topfirst bottom)\log_{2}\!\left(\frac{\text{last top}}{\text{first bottom}}\right).

Establish the structural invariant and eliminate impossible options immediately.

STEP 2 • FAST SOLVE (LINE 2)15–25s
Line 2 (Evaluate Power): log⁡2 ⁣(483)=log⁡2(16)=4\log_{2}\!\left(\frac{48}{3}\right) = \log_{2}(16) = 4 (20s).

Evaluate the simplified expression using integer factoring or unit elimination.

Distractor Trap Forensics

The predictable cognitive traps engineered by examiners into the wrong options.

TRAP #1fence_post_or_domain_trap

Accepting roots where u = aˣ <= 0 or log argument x <= 0

Failing to reject non-positive roots when solving quadratic in u = aˣ or combining log(x-a) + log(x-b).

Sub-90 Fix: Test invariants before expanding equations.
TRAP #2sign_or_direction_flip

Failing to flip inequality sign when dividing by logₐ(b) where 0 < b < 1 (since logₐ(b) < 0)

Treating log(0.5) as a positive number during inequality division.

Sub-90 Fix: Test invariants before expanding equations.
TRAP #3factor_of_two_or_stoichiometry_trap

Confusing log(A) + log(B) = log(A+B) instead of log(AB), or (log x)² with 2*log(x)

Applying linear distribution to logarithmic laws.

Sub-90 Fix: Test invariants before expanding equations.

Try 1 Timed Question: Laws of Logarithms, Disguised Exponential Quadratics (a²ˣ + b aˣ + c = 0) & Log Inequalities

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Full Lesson All 15 Questions (ARCH-MM-07)
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ARCH-MM-07: Laws of Logarithms, Disguised Exponential Quadratics (a²ˣ + b aˣ + c = 0) & Log Inequalities (Sample)Mathematics 2
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SUB-90 ZONE (<35S)
0.0s35s Sub-9090s ESAT Budget150s Classroom
Question 1 of 1ARCH-MM-07 (6-Card Lesson)Format: Standard 5-Option
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Question 1 of 1 • 90s Target

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