$(24$ points$)$ For each pair of expressions $(A,B)$ below, indicate whether $A$

is $O,,$ or $$ of $B.$ Note that zero, one, or more of these relations may

hold for a given pair. List all applicable relations. No explanation

is needed. If $A=(B)$ then write $A=(B)$ which already implies

that $A=O(B)$ and $A=(B).$ If $A=(B)$ and you write $A=O(B)$

or $A=(B)$ only, you will only receive partial credits. It often happens

that some students will get the directions wrong, so please write out the

relation in full, i$.$e$.,A=O(B),A=(B),$ or $A=(B)$ and not just

$O(B),(B)$ or $(B).$

$($a$)A=n^{2}logn+50n^2,B=\frac{n^3}{logn}+3n^{2\mathrm{.}9}$;

$($b$)A=lo{g}_3(n^2),B=lo{g}_{10}(n^4)$;

$($c$)A=lo{g}_5(n^{n^2}),B=lo{g}_2(n^n)$;

$($d$)A=2^n,B=n^{logn}+8n^2$;

$($e$)A=4^{3n},B=5^{2n}$;

$($f$)A=3^{lo{g}_{2}n},B=n^{lo{g}_{2}3}.$