3. Given positive real numbers $a(a \neq 1) 、 x 、 y$ satisfy
$$
\log _{a}^{2} x+\log _{a}^{2} y-\log _{a}(x y)^{2} \leqslant 2,
$$

and $\log _{a} y \geqslant 1$. Then the range of $\log _{a} x^{2} y$ is
3. $[-1,2 \sqrt{5}+3]$.

Let $u=\log _{a} x, v=\log _{a} y$. Then $z=\log _{a} x^{2} y=2 \log _{a} x+\log _{a} y=2 u+v$. From the given conditions, we have
$$
\left\{\begin{array}{l}
(u-1)^{2}+(v-1)^{2} \leqslant 4, \\
v \geqslant 1 .
\end{array}\right.
$$

Then the point $(u, v)$ corresponds to the shaded region in Figure 1. By $v=-2 u+z$, we know that $z$ is the intercept on the $v$-axis. Therefore, $-1 \leqslant z \leqslant 2 \sqrt{5}+3$.