I want to approximate a real number $r$ as a logarithm using integer numbers: \begin{equation} r\approx\log_a K \;(a,K \in \Bbb N,\; r\in \Bbb R) \end{equation} Intuitively, the larger $a$ and $K$ are allowed to be, the better the approximation.
Of course, I could seach numerically the best approximation (e.g. with a computer program), but I imagine there must be an algorithm to directly find a list of progressively better approximations. For example, if instead of a logarithm I were using a fraction $r=a/K$, I could use the method of continued fractions.
Example: for $r=1.3807$, with a program I found $r \approx \log_5 9,\;\log_{6} 12,\;\log_9{21},\;\log_{10}24,\;\dots$
How do I find the next pair $(a,K)$?