An exponential equation

Substitution. $t=2^x>0$, so $4^x=t^2$. The equation becomes
$$t^2-5t+4=0,$$
i.e. $(t-1)(t-4)=0$, so $t=1$ or $t=4$.

Back. $2^x=1\Rightarrow x=0$; $2^x=4\Rightarrow x=2$.

Conclusion. $x\in\{0,2\}$.