\section*{Exercise 2 - 081022}

Determine all real numbers \(x\) that satisfy the condition \(\log _{2}\left[\log _{2}\left(\log _{2} x\right)\right]=0\)!
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If \(x\) satisfies the required condition, we set \(\log _{2}\left(\log _{2} x\right)=a\) and obtain \(\log _{2} a=0\).

This means \(2^{0}=a\), so \(a=1\).

Thus, \(\log _{2}\left(\log _{2} x\right)=1\). We set \(\log _{2} x=b\) and obtain \(\log _{2} b=1\). This means \(b=2\).

Thus, \(\log _{2} x=2\), so \(x=4\) if it is to satisfy the required condition.

Solution adopted from \([5]\)