Suppose you have an algorithm with time complexity $T(n)=((lo{g}_2(n){)}^3)$

which for inputs of size $n=1000$ runs in exactly $1$ second on your computer.

Which of the following alternatives is the most reasonable guesstimate for how

large values of $n$ the algorithm could solve in $1$ minute $(60$ seconds$)?$

$($Even though the nature of asymptotics are such that you cannot know exactly

how long the algorithm will run, you should give a reasonable best guess based

on the available information. Assume that the time complexity $T(n)$ is the only

thing you know about the algorithm and that e$.$g$.$ its memory usage does not

have any strange effects on its running time.$)$

$n=500$

$n=1050$

$n=1500$

$n=7000$

$n=25000$

$n={10}^5$

$n={10}^{10}$

$n={10}^{20}$

Can you explain the solution?