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.

Starting with $12^n < 2^m < 13^n$, we see that $n \log_2(12) < m < n \log_2(13)$. We conclude that 
\begin{align}
   m &= \left\lceil n\dfrac{\log(12)}{\log(2)} \right\rceil \\
     &\approx \lceil n \times 
     3.584962500721156181453738943947816508759814407692481060455\dots
     \rceil
\end{align}
The first few values of $m$ and $n$ are
$(m,n) 
\in \{(11, 3), (15, 4), (18, 5), (22, 6), (26, 7), (29, 8), (33, 9), (36, 10)\}$
You can see that $m=11$ and $n=3$ will be the smallest values of $m$ and $n$.
Note also that there is no smallest value of $\dfrac mn$ since 
$\displaystyle \lim_{n \to \infty} \dfrac mn = \log_2 12$ and $\log_2 12$ is irrational.
