You can do it by setting up a truth table, if you do that correctly then you will find out that they are logically equivalent. However with such an easy example you can do it with a few small steps of reasoning:
Take $A \vee \neg B $, we can see that with the disjunction the only way to get $0$ is to have $A \equiv 0$ and $B \equiv 1$, in all other cases it's $1$.
Take $\neg A \rightarrow \neg B $, we know that a implication is only $0$ if we have $1 \rightarrow 0$, but with the negations in our example, we can see that it's only $0$ if we have $0 \rightarrow 1$ (i.e. the negation of $1 \rightarrow 0)$
We can see that in both steps we get that the expression is only $0$ iff $A \equiv 0$ and $B \equiv 1$. This must then mean that they are logically equivalent (all other cases for both expressions become $1$ all around).