ARCH-MM-07: Laws of Logarithms, Disguised Exponential Quadratics (a^{2x} + b a^x + c = 0) & Log Inequalities

Full 15-Q Timed Drill

Step 1 of 6 • The Core Idea

ARCH-MM-07: Why the 150s Classroom Method Fails Under the 90s Clock

Standard Classroom Method (Too Slow)~150s (Blows 90s Clock)

Using

\log_{2}(A) + \log_{2}(B) = \log_{2}(AB)

, the sum becomes

\log_{2}\!\left(\frac{4}{3} \times \frac{5}{4} \times \dots \times \frac{48}{47}\right) = \log_{2}\!\left(\frac{48}{3}\right) = \log_{2}(16) = 4

(~90s).

The Fast ShortcutTarget: 40s (Saves 110s)

Line 1 (Telescoping Log Product): All interior numerators and denominators cancel, leaving

\log_{2}\!\left(\frac{\text{last top}}{\text{first bottom}}\right)

.
Line 2 (Evaluate Power):

\log_{2}\!\left(\frac{48}{3}\right) = \log_{2}(16) = 4

(20s).

Where Students Lose Time: In 47 past UAT-UK questions (11.46% of Mathematics 2), candidates lose 110s by failing to reject non-positive roots when solving quadratic in u = a^x or combining log(x-a) + log(x-b). Instead, apply MM5 with Maths 1 skill M2.6.

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