Existence problem

Let us define the region $S = [(x,y) | x \in [a,2a], y \in [a, a^2], a > 1]$ in the $xy-$ plane. For $c \in \mathbb{R}$ , let $log_{a}x + log_{a}y = c \Rightarrow xy = a^{c} \Rightarrow y = \frac{a^c}{x}$ (i), which is a decreasing function over $a \le x \le 2a.$ Taking the point $(x,y) = (a, a^2)$ , we find that:

$a^2 = \frac{a^c}{a} \Rightarrow c = 3$ (ii).

At the point $(x,y) = (2a,a)$ we obtain $a = \frac{a^3}{2a} \Rightarrow a^3 - 2a^2 = a^2(a-2) = 0 \Rightarrow a = 0, 2.$ Since we are given that $a > 1$ , the only such value is $\boxed{a=2}.$