Problen $6.\backslash ((15+15=30\backslash )$ points$)$ Resul Sertion $10\mathrm{.}6$ carefolls bufore uthenting this problem. Analyge the puming time of the following algorithu using a step count analysis as showa in the Homer scheme $($Example $10$:$40).$

$```$

$//$ determine the number of digits of an integer n

binary$\_$digits$($n$)$ cost times

int ont $=1$ cl $[]$

Wijile $($ii $>1)$ do $$ ic $]$

cnt $=$ cnt $+1$ c$3[]$

n $=$ floor$($ n$/2.$Q$)$ c$4[]$

endwhile

return ent

c$5$: $[]]$

c$6[]$

$```$

$($a$)$ Fitl im the $[]$s in the above code each with a nomber or an expression involving a that expresses tho step count for the line uf code.

$($b$)$ Detenume the worst$-$ase emppexity of this algorithm as a function of $\backslash ($ n $\backslash )$ and give it in the $\backslash (\backslash$Theta $\backslash )$ notation. Show your work and explain using the defintion of $\backslash (\backslash$Theta $\backslash )$ involying ther irecgualities.