You may let x be fixed and let g(y)=y*ln(x)-x*ln(y).
The first step, you to show that g(x)=0, (say, let y=x)
2nd, g'(y)=ln(x)-x/y, and we have g'(x)=ln(x)-1>0 (since x>3);
3rd, g''(y)=x/y^2>0, so g'(y) is monotonically increasing in y>x>e, and therefore, g'(y)>0 for y>x>e.
and...