Task B-3.7. Let $a \in \mathbb{R}, a>0, a \neq 1$. Determine all $x \in \mathbb{R}, x>1$ for which the expression

$$
x^{2-\log _{a}^{2} x-\log _{a}\left(x^{2}\right)}-\frac{1}{x}
$$

is positive.
## Solution.

$$
\begin{aligned}
& x^{2-\log _{a}^{2} x-\log _{a}\left(x^{2}\right)}-\frac{1}{x}>0 \\
& x^{2-\log _{a}^{2} x-\log _{a}\left(x^{2}\right)}>\frac{1}{x}
\end{aligned}
$$

Since the base $x>1$, it follows that

$$
2-\log _{a}^{2} x-2 \log _{a} x>-1
$$

We obtain the inequality

$$
\log _{a}^{2} x+2 \log _{a} x-3<0
$$

Solving this quadratic inequality, we get

$$
-3 < \log _{a} x < 1
$$

1) For $a>1$, since $a^{-3} < 1$, we have $x \in (1, a)$. (2 points)

2) For $0 < a < 1$, since $a^{-3} > 1$, we have $x > a^{-3}$. Together with $x > 1$, we get $x \in (1, a^{-3})$. (2 points)