jmc

We have $\frac{1}{4}\cdot 2^{30}=\frac{2^{30}}{2^2}=2^{30-2}=2^{28}$. We also have $4^x=(2^2{)}^x=2^{2x}$. Setting these equal gives $2^{28}=2^{2x}$, so $2x=28$, which means $x=\boxed{14}$.