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:
$8n\log(2n) > n^2$
Thank you for your help.
Since $n>0$, divide both sides and consider the function $$f(x)=8\log(2x)-x$$ 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 $1 \leq n \leq 14$.