Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 4.7 AIME Find the answer

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

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

We compute $N^{(N^{N})}=16^{16^{16}}=2^{4 \cdot 2^{4 \cdot 2^{4}}}=2^{2^{2^{6}+2}}=2^{2^{66}}$ so $x=66$ .

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