HMMT-Feb 2019 alg, Problem 2

Let $N=2^{\left(2^{2}\right)}$ and $x$ be a real number such that $N^{\left(N^{N}\right)}=2^{\left(2^{x}\right)}$. Find $x$.

Solution: We compute \[ N^{\left(N^{N}\right)}=16^{16^{16}}=2^{4 \cdot 2^{4 \cdot 2^{4}}}=2^{2^{2^{6}+2}}=2^{2^{66}} \] so $x=66$. $\fbox{66}$.

Answer: 66