$$4^{2(x-2)}=16$$
$x=3$
$$4^{2\left(x-2\right)}=16$$
$$\log(4^{2\left(x-2\right)})=\log(16)$$
$$2\left(x-2\right)\log(4)=\log(16)$$
$$2\left(x-2\right)=\frac{\log(16)}{\log(4)}$$
$$2\left(x-2\right)=\log_{4}\left(16\right)$$
$$x-2=\frac{2}{2}$$
$$x=1-\left(-2\right)$$
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