TMUA 2017 · Paper 2 · Question 18

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:

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

$↓$ (I)

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

$↓$ (II)

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

$↓$ (III)

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