3-7 Iterated functions

We can apply the iteration operator * used in the $\lg^{\ast}$ function to any monotonically increasing function $f(n)$ over the real. For a given constant $c \in \R$, we define the iterated dunction $f_{c}^{\ast}$ by

$f_{c}^{\ast}=\min \lbrace i\geq0:f^{(i)}\leq c\rbrace$,

which need not be well defined in all cases. In other words, the quantity $f_{c}^{\ast}(n)$ is the minimum number of iterated applications of the function $f$ required to reduce its argument down to $c$ or less.

For each of the functions $f(n)$ and constants $c$ in the table below, give as tight a bound as possible on $f_{c}^{\ast}(n)$. If there is no $i$ such that $f^{(i)} \leq c$, write "undefined" as your answer.

$f(n)$	$c$	$f_{c}^{\ast}(n)$
$n-1$	$0$	$\Theta(n)$
$\lg n$	$1$	$\Theta(\lg^{\ast}n)$
$n/2$	$1$	$\Theta(\lg n)$
$n/2$	$2$	$\Theta(\lg n)$
$\sqrt{n}$	$2$	$\theta(\lg\lg n)$
$\sqrt{n}$	$1$	undefined
$n^{1/3}$	$2$	$\Theta(\log_3\lg n)$