Answer
$\forall x(P(x)\rightarrow Q(x))$ and $\forall xP(x)\rightarrow \forall xQ(x)$ are not logically equivalent.
Work Step by Step
Assume that the domain contain y and z for which P(y) is false, Q(y) is false, P(z) is true and Q(z) is false.
Then, $\forall x(P(x)\rightarrow Q(x))$ is false (since for x=z, P(z) is true but Q(z) is false) but $\forall xP(x)\rightarrow \forall xQ(x)$ is true (since $\forall x P(x) $ is false).
Hence, $\forall x(P(x)\rightarrow Q(x))$ and $\forall xP(x)\rightarrow \forall xQ(x)$ are not logically equivalent.
