Closest power to another power

Question

I am interested in finding the closest power of one number to another. Specifically, if we have $k,m,n\in\mathbb{N}$, $\gcd(m,n)=1$, and $m<n$, how can we minimize $n^k-m^{\lfloor\log_m(n^k)\rfloor}$? Heuristically, as k increases so does the difference, but I cannot seem to demonstrate this conclusively.