Answer
N = 1383
Work Step by Step
We can graph the function $f(x) = \sqrt{x~ln~x}$
On the graph , we can see that $~~f(x) \gt 100~~$ when $~~x \gt 1383$
Therefore:
If $x \gt 1383,~~~$ then $~~~ \sqrt{x~ln~x} \gt 100$
We can check the value $N=1383$ in the function:
$f(1383) = \sqrt{(1383)~(ln~1383)} = 100.009$
