  Combining the ideas of Riesz $s$-energy and $\log$-energy, we introduce the
so-called $s,\log^t$-energy. In this paper, we investigate the asymptotic
behaviors for $N,t$ fixed and $s$ varying of minimal $N$-point
$s,\log^t$-energy constants and configurations of an infinite compact metric
space of diameter less than $1$. In particular, we study certain continuity and
differentiability properties of minimal $N$-point $s,\log^t$-energy constants
in the variable $s$ and we show that in the limits as $s\rightarrow \infty$ and
as $s\rightarrow s_0>0,$ minimal $N$-point $s,\log^t$-energy configurations
tend to an $N$-point best-packing configuration and a minimal $N$-point
$s_0,\log^t$-energy configuration, respectively. Furthermore, the optimality of
$N$ distinct equally spaced points on circles in $\mathbb{R}^2$ for some
certain $s,\log^t$ energy problems was proved.
