Let $\displaystyle a_n=\int_0^n\int_0^{\frac{1}{n}}\int_0^n\int_0^{\frac{1}{n}}\log^2\Big[(x-t)^2+(y-s)^2\Big]dx\,dy\,dt\,ds,$ for $n=1,2,\cdots$.
How to show that $\displaystyle\lim_{n\uparrow\infty}a_n=+\infty?$ (A Wolfram Mathematica experiment confirms that $a_n\to+\infty$)
The integrand is nonnegative, so the integral is at least as large as what we get integrating over a smaller region of $4$-space. Let's look at the region $2n/3 < y < n, 0<s<n/3,$ $0 < x,t < 1/n.$ In this region, for large $n,$ the integrand is at least $\ln^2 (n/3)^2.$ The four dimensional volume of this region is $(n/3)^2\cdot (1/n)^2 = 1/9.$ Thus the integral is at least $\ln^2 (n/3)^2\cdot (1/9) \to \infty.$