why the largest value of $k$ for any number of the form $^<B$ is $\lfloor\log B \rfloor$

Question

Given an integer $n>2$, let $B=\lceil\log^5n\rceil$. I am trying to understand why the largest value of $k$ for any number of the form $m^k \le B$, $m\ge2$ is $\lfloor\log B \rfloor$. Logarithms are in base $2$. It seems easy but I can't prove it. Any hint would be precious, thanks a lot!