About the representation of any integer.

Question

Today I was going through Yakov Perelman's book Mathematics can be fun, I found without proof that "Every natural number can be represented only by using $2$s and mathematic symbols".

For example $3 = - \log_2 \log_2\sqrt{\sqrt{\sqrt{2}}}$. But can someone please provide a proof of why this happens? I searched in MSE but found no related topics. It can be generalized as $$N = - \log_2 \log_2\sqrt{\sqrt \dotsi (\text{n times)}\dotsi{\sqrt{2}}}~.$$ Thanks!

Answer 1

$$\overbrace{\sqrt{\sqrt{\sqrt{\dots\sqrt{2}}}}}^{n\text{ times}}=2^{1/2^n}$$ $$\log_2{\left(\overbrace{\sqrt{\sqrt{\sqrt{\dots\sqrt{2}}}}}^{n\text{ times}}\right)}=\log_2{(2^{1/2^n})}=\frac1{2^n}=2^{-n}$$ $$\therefore-\log_2\log_2{\left(\overbrace{\sqrt{\sqrt{\sqrt{\dots\sqrt{2}}}}}^{n\text{ times}}\right)}=-\log_2{\left(2^{-n}\right)}=-(-n)=n$$

Answer 2

Because $\Large{\sqrt{\sqrt..(\text{n times)}..{\sqrt{2}}}=2^{\frac{1}{2^n}}}$

and $\log_2{2^{\frac{1}{2^n}}}=\frac{1}{2^n}=2^{-n}$