Identifying when the functi0n $\log(x!)-k\log((\frac{x}{2})!)$ is strictly increasing

Question

Let $x$ and $k$ be natural numbers. For which $k$ natural numbers the function $\log(x!)-k\log((\frac{x}{2})!)$ is strictly increasing?

Answer 1

HINT: Have you tried to shift the expression to $\log\left(\frac{x!}{(\frac{x}{2})!^k}\right)$? Because $\log$ is strictly monotone its enough to study $\frac{x!}{(\frac{x}{2})!^k}$