In this problem it will be useful to recall the following properties of logarithms: log$($xy$)=$log$($x$)+$log$($y$)$ and log$($xa$)=$alog$($x$).$

Find the least integer k such that f$($n$)$ is O$($nk$)$ for each of the following functions:

$($a$)$ f$($n$)=$nlog$(4$n$)2$

$($b$)$ f$($n$)=12+22++$n$23$

$($c$)$ f$($n$)=$log$($n$!)2$

$($d$)$ f$($n$)=$log$($nn$)$n$9+1$