  We provide logarithmic gradient estimate and universal boundedness estimate
for semilinear elliptic equations on RCD$^*(K,N)$, metric measure spaces. In
certain case, these estimates are optimal even on RCD$^*(K,N)$ spaces with
$K<0$. Two direct corollaries of these estimates are Harnack inequality and
Liouville theorem. In addition to these estimates, we also establish relations
among the universal boundedness estimate, the logarithmic gradient estimate,
and Harnack inequality. Under certain conditions, even in wild setting, they
are $\kappa$-equivalent on RCD$^*(0,N)$ spaces for any $\kappa>1$.
