I was reading a research paper and there I read the following:
$t=O\left(d^{2} \log _{d}^{2} n\right)$ matches the lower bound $\Omega\left(d^{2} \log _{d} n\right)$ in the regime where $d=\Theta\left(n^{\alpha}\right)$ for some $\alpha \in(0,1)$.
It also says $t=O\left(d^{2} \log n\right)$ outperforms $t=O\left(d^{2} \log _{d}^{2} n\right)$ in the regime where $d=O(\operatorname{poly}(\log n))$
I am not sure how to verify these statements? I understand the basics of all the complexity notations, but how can we mathematically prove this?