Log reduce PATH to DISTANCE-PATH
The idea is to build a layer graph $G'=\left (V_1,...,V_n\right)$ with $n$ layers, where each layer is a copy of $V$. For each $(u,v)\in E$ we add the edges $\left\{(u_i,v_{i+1}) | 1\le i \le n-1\right\}$, where $u_i$ is the copy of $u$ in $V_i$. In addition we add self edges $\left\{(u_i,u_{i+1}) | u\in V, 1\le i\le n-1\right\}$. You then want to know whether there exists a path $s_1\rightsquigarrow t_n$. Since $G'$ is a layer graph, any path from $s_1$ to $t_n$ is of length $n-1$ (so you dont need to worry about shorter paths anymore). You should now be able to complete the details yourself.