$(\log_23)^x-(\log_53)^x\geq(\log_23)^{-y}-(\log_53)^{-y}$
I guess the function $f(x)=(\log_23)^x-(\log_53)^x$ monotonically increasing, so I get the answer $x\geq-y$,but how to prove it not using calculus?
Or how to proof $f(x)=a^x-b^x(a>b)$ is monotonically increasing?