For relatively small values of $n,$ algorithms with larger orders can be more efficient than algorithms with smaller orders.

Suppose that when run with an input of size $n,$ algorithm A requires ${10}^7{n}^4$ operations and algorithm $B$ requires $n^5$ operations.

$($a$)$ What are orders for algorithms A and $B$ from among the set of power functions?

algorithm A order

algorithm $B$ order

$r$

$($b$)$ For what values of $n$ is algorithm A more efficient than algorithm B$?($Enter your answer as a single inequality solved for $n.)$

$($c$)$ For what values of $n$ is algorithm $B$ at least $100$ times more efficient than algorithm $A?($Enter your answer as a single inequality solved for $n.)$