[Algebra]
Problem: Define $\log ^{*}(n)$ to be the smallest number of times the log function must be iteratively applied to $n$ to get a result less than or equal to 1 . For example, $\log ^{*}(1000)=2$ since $\log 1000=3$ and $\log (\log 1000)=\log 3=0.477 \ldots \leq 1$. Let $a$ be the smallest integer such that $\log ^{*}(a)=3$. Compute the number of zeros in the base 10 representation of $a$.

Solution: If $\log ^{*}(a)=3$, then $\log (\log (\log (a))) \leq 1$ and $\log (\log (a))>1$. If $\log (\log (a))>1$, then $\log (a)>10$ and $a>10^{10}$. Because the problem asks for the smallest such $a$ that is an integer, choose $a=10^{10}+1=10,000,000,001$, which has 9 zeros.

Final answer: 9