Provide tight asymptotic bounds $($i$.$e$.(*)$ on the following. Correct answers are given full

points without explanation. $(4$ points each for the first three parts, $5$ points for the last part$)$

$($a$)l{g}^3(n)+5^{lglgn}+l{g}^2(l{g}^{4}n)$

$($b$)$ The number of bits needed to write ${12}^n$ in binary.

$($c$)$ Planet $x$ has a current population of $1$ billion. Assuming that the population of planet

$x$ doubles every year, give an asymptotic bound on the population of planet X in $100$

years from now.

$($d$)$ Give two functions, $f,g$:$NN$ such that $f(n)$ is not $O(g(n))$ and $g(n)$ is not $O(f(n)).$