Sort the following $25$ functions from asymptotically smallest to asymptotically largest, indi$-$

cating ties if there are any. You do not need to turn in proofs $($in fact, please don't turn in

proofs$).$

To simplify notation put all the functions in a single column, where $f(n)$ being above

$g(n)$ in the column means $f(n)=o(g(n)).$ If $f(n)=(g(n)),$ then write $f(n)$ and $g(n)$

separated by a comma on the same line. For example, the functions $n^2,n,([n],[2]),n^3$ should be

written

$n$

$n^2,([n],[2])$

$n^3$

Hint: When considering two functions $f(n)$ and $g(n)$ it is sometimes useful to consider

the functions $logf(n)$ and $logg(n),$ since $logf(n)=o(logg(n))$ implies $f(n)=o(g(n))$

$($but be careful, the implication does not work in the other direction, and in particular

$logf(n)=O(logg(n))$ does NOT imply $f(n)=O(g(n)).$