jmc

We are asked to solve $4^x=8$ for $x$. Writing $4$ as $2^2$ and $8$ as $2^3$, the equation becomes $(2^2{)}^x=2^3$. The left-hand side simplifies to $2^{2x}$, so we may set exponents equal to find $2x=3$, which implies $x=\boxed{\frac{3}{2}}$.