Solving inequality including logarithm

Question

I am trying to solve an inequality which includes a logarithm. This is to show for what input one algorithms is faster than another. I do not know how to change the inequality so that I can solve it. The problem is as follows:
$8nlog(2n) > n^2$
Thank you for your help.

Claude Leibovici · Answer

Since $n>0$, divide both sides and consider the function
$$f(x)=8log(2x)-x$$ for which
$$f'(x)=frac{8}{x}-1 qquad text{and} qquad f''(x)=-frac{8}{x^2}<0 quad forall x$$ The first derivative cancels at $x=8$ and by the second derivative test, this is a maximum.
Since $f(8)=32 log (2)-8 sim 14.18$, then, if $n$ is a natural number, the inequality holds for $1 leq n leq 14$.

Solving inequality including logarithm

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