Prove using logarithms

Question

I don't know how to prove that:

$$(2n-1)^{2n-1} \geq n^{2n} \mid \forall n\in \mathbb N$$

I have to use logarithms. I have tried to transform it into the form: $\log_n(2 - 1/n) \geq \frac{1}{2n-1}$, but I think I'm wrong and don't know what to do further.

Answer 1

Let $f(x)=(2x-1)\ln(2x-1)-2x\ln{x},$ where $x\geq1$,

Thus, $$f'(x)=2(\ln(2x-1)-\ln{x})\geq0,$$ which says $f(x)\geq f(1)=0.$