So I came across with this problem: To prove $\log_27<2\sqrt3$. This is not difficult as LHS$<3<$RHS.
However, I did want to know how to generalise the solution. In particular, given integers $M,N$, how can we prove a rational number $r=\frac pq$ that is greater than $\log_MN\,?$
An attempt I tried is to generate a sequence of rationals that converges to $\log_MN$, however, that did not go very well for me.
Thank you.
We have $$ \log_MN<\frac{p}{q} \iff N^q<M^p. $$ So, the question is equivalent to computing powers of integers.
Ex.
$$ 8=2^3<3^2=9 \implies \log_{3}{2}<\frac{2}{3} $$
One can get rational approximations of $\log_MN$ by a variant of Euclid's algorithm. Start with $M$ and $N$ and repeatedly divide the larger number by the smaller one until the numbers become close to $1$. This gives something of the form $M^pN^{-q}\approx 1$. Then $\log_MN\approx \frac{p}{q}$.
Ex.
\begin{align*} 2,\quad &3\\ 3/2,\quad &2\\ 3/2,\quad &2^2/3\\ 3^2/2^3,\quad &2^2/3\\ 3^2/2^3,\quad &2^8/3^5\\ \vdots \end{align*}
and these give rational approximations to $\log_32$
\begin{align*} 0,\quad &1/0\\ 1,\quad &0\\ 1,\quad &1/2\\ 2/3,\quad &1/2\\ 2/3,\quad &5/8\\ \vdots \end{align*} The last approximation $0.625$ is close to the true value $0.6309$.