TMUA 2017 Paper 2 Q18 — Worked Solution

P2·Indices, Logarithms & Exponentials, Inequalities·6.5Official · 2017 · Q18

proof-errorlog-base-less-than-oneinequality-direction

Consider the following problem: Solve the inequality

(\frac{1}{4})^{n} < (\frac{1}{32})^{10}

, where

n

is a positive integer.A student produces the following argument. Starting from

(\frac{1}{4})^{n} < (\frac{1}{32})^{10}

, the steps (I) through (V) lead to:

(I)

lo g_{1/2} (\frac{1}{4})^{n} < lo g_{1/2} (\frac{1}{32})^{10}

(II)

n lo g_{1/2} (\frac{1}{4}) < 10 lo g_{1/2} (\frac{1}{32})

(III)

n < \frac{10 lo g _{1/2} ( 1/32 )}{lo g _{1/2} ( 1/4 )}

(IV)

n < \frac{10 \times 5}{2} = 25

(V)

1 \leq n \leq 24

Which step (if any) in the argument is invalid?

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