Consider $V = \lbrace \log(n) : n \in \mathbb N_{>0} \rbrace$ and scalars $s \in \mathbb Z_q$ for $q $ prime. Elementary number theory tells you that every $\log(n)$ can be written in a unique way as $s_1 \log(p_1) + \dots + s_k \log(p_k)$ for certain primes $p_1 \dots p_k$. One is tempted to think of $\lbrace \log(p) : p \, \text {prime} \rbrace$ as a basis, but, of course, $s_i \in \mathbb N$, the set of scalars, is not a field. Restricting those scalars to be $\mathbb Z_q$ may give you a field but then the corresponding subset of $V$ with standard addition does not play well with the field’s inverse for addition, etc.
Is there any theory being developed around this simple idea, albeit with modifications, to make it work in some restricted sense? I mean, the idea of seeing $\log(p)$ with $p$ prime as a basis for $\log(n)$ as an (almost) vector space?
