How to verify Hamilton-Path in log-space?
You can check all vertices in order.

Suppose we are given vertices $v_1, v_2, \cdots, v_n$ and a path $p=(p_1,p_2, \cdots,p_m)$.Check if $m==n$. If not, reject. Otherwise, check if $v_1$ appears in $p$. If not, reject.  Otherwise, check if $v_2$ appears in $p$. If not reject. And so on. Finally check if $v_m$ appears in $p$. If not, reject. Otherwise, accept.