Example 3 Find the minimum value of the function $\sqrt{2 x^{2}-4 x+4}+\sqrt{2 x^{2}-16 x+\log _{2}^{2} x-2 x \log _{2} x+2 \log _{2} x+50}$.
When faced with such a complex radical, the first thing that comes to mind is the distance formula between two points in analytic geometry.
Solution Consider three points $A(0,0)$, $B(x, x-2)$, and $C\left(7, \log _{2} x-1\right)$ in a Cartesian coordinate system, then we can derive $y=|A B|+|B C| \geqslant|A C|$ (since the shortest distance between two points is a straight line), hence $y \geqslant$ $|A C|=\sqrt{49+\left(\log _{2} x-1\right)^{2}} \geqslant 7$, and there exists $x_{0}=2$ such that $y=7$. Therefore, the minimum value of $y$ is 7.