jmc

We can write $$16^{(2^x)}=(2^4{)}^{(2^x)}=2^{4\cdot 2^x}.$$Then $2^{16^x}=2^{4\cdot 2^x},$ so $$16^x=4\cdot 2^x.$$In turn, we can write this as $$2^{4x}=2^{x+2},$$so $4x=x+2.$ Therefore, $x=\boxed{\frac{2}{3}}.$