For any natural number n $>0,$ log$$ n is defined recursively as follows:

log$$ n $=$

$$

$1+$ log $($log n$)$ if n $>1$

$0$ if n

Show that log$$ n $=$ o$($log log n$)$ and log$$ n $=\backslash$omega $(1).$Question $2(4$ points$)$ For any natural number $n>0,lo{g}^{***}n$ is defined recursively as follows:

$lo{g}^{***}n=\{\begin{array}{ccc}1+log**(logn)& if& n>1\\ 0& if& n1\end{array}$

Show that $lo{g}^{***}n=o(loglogn)$ and $lo{g}^{***}n=(1).$