Cryptographic procedures seem to almost exclusively use discrete logarithms rather than continuous logarithms. Hence, I assume there are good and sound reasons for this.
In essence, answers provided here: Why is NON DISCRETE logarithm problem not hard as the DISCRETE logarithm problem (so computationally hard)? state that continuous logarithms, thus, non-discrete logarithms can be computationally fast, while for discrete logarithms no such algorithm is known.
Furthermore, continuous logs can pose the problem of mathematical precision in terms of rounding errors.
For any non-discrete logarithm problem such as:
$$v = \log_{b}e$$
solving for either an unknown base $b$ or an unknown exponent $e$ is trivial when $v$ is known.
So my question is, not considering rounding issues, for now, what about solving for $v$ in the equation above, however when both the base and exponent are unknown, with the base $b$ being prime and $e$ being a random integer greater than 1.