Is it possible to get nontrivial $n$ such that we can find BOTH $n \pmod p$ and $\log{(n)} \pmod p$?

Question

If we are looking for a value $n \equiv v \pmod p$ or $n \equiv v_r + v_i i \pmod p$, where $v_r+v_i\cdot i$ is a complex number modulo $p$, is it ever possible to have a situation where we can find both $n$ and $\log{(n)}$ modulo $p$? I know that, for instance, if $n=1$, then $\log{(n)} = 0$. However, I'm looking for a situation in which both values are nonzero.