For $1>x>y>0$ and $n\geq 1$, prove $$\left(\frac{x}{y}\right)^n \geq \frac{\ln y}{\ln x}.$$
Attempt: Rewrote it as $$x^n \ln(x)- y^n \ln y\leq 0$$
Tried to show that the LHS is quasi-convex in $n$ and that it attains its lowest value for $n\leq 1$.
Got a bit more complicated.