Smallest Possible Power When working on improving my skills with indices, I came across the following question:

Find the smallest positive integers $m$ and $n$ for which: $12<2^{m/n}<13$

On my first attempt, I split this into two parts and then using logarithms found the two values $m/n$ had to be between. However I wasn't sure how to progress past that.
I have the answer itself $(11/3)$, but I'm unsure of the best method to find it. Any help would be really appreciated.

The inequality is equivalent to $\,12^n \lt 2^m \lt 13^n\,$. By brute force, looking for powers of $2$ between $12^n$ and $13^n$ starting from the lowest possible $n=1$ up:


*

*$\;n=1\,$: no solutions, since $\,2^3 = 8 \lt 12^1 \lt 13^1 \lt 16=2^4\,$

*$\;n=2\,$: no solutions, since $\,2^7 = 128 \lt 144 = 12^2 \lt 13^2 = 169 \lt 256=2^8\,$

*$\;n=3\,$: $\,12^3 = 1728 \lt 2048 = 2^{11} \lt 2197 = 13^3\,$, therefore $m=11, n=3$ is a solution.
