Solving continuous logarithm whith unknown base and unknown exponent

Question

I was wondering if there is any feasible solution for finding the base and exponent of a continuous logarithm when only the result is known. Thus:

$v=\log_{b}e$

with only $v$ given and $b,e$ are integers greater 1. By feasible I mean an approach other/faster than brute-force.

Answer 1

Not really, as there are infinite solutions if both $b$ and $e$ are undetermined. Consider, as an example:

$2= \log_{10}100$, but also $2= \log_{3}9$

In general, $v= \log_{b}b^v$

Answer 2

Using natural logarithms (the only ones I know) $$v=\log_{b}(e) \implies v=\frac{\log (e)}{\log (b)}$$ So, if only $v$ is known, you have an infinite number of possible combinations of $(b,e)$.