Inspired by the question on which points of the $\mathbb{C}$ unit circle can be reached by arbitrary products of two example points, I came up wtih the following:
For given $a, b \in \mathbb{R}^+ \backslash \{1\}, a \neq b$, a target value $c \in \mathbb{R}_0^+$ and $\epsilon > 0$, find $k,l \in \mathbb{Z}$ such that $|c - a^k \cdot b^l| < \epsilon$.
This is obviously prone to failure with inappropriate "generator" choices like $a = b^m, m \in \mathbb{Z}$. Now I am wondering under which conditions such an approximation is possible? And how would you compute $k,l$? Searching integer points $(k,l)$ closest to the line $\tilde{k} \cdot \log(a) + \tilde{l} \cdot \log(b) = \log(c)$ would be my first strategy.
An upgrade to the question would be:
Find a sequence $\mathbb{N} \rightarrow \mathbb{Z}^2, n \mapsto (k_n, l_n)$ such that $\lim_{n \rightarrow \infty} a^{k_n} \cdot b^{l_n} = c$.