What should I study to further explore this approach to the arithmetic derivative?

So I am studying the arithmetic derivative and I stumbled upon following approach: We are considering here a $\Bbb{Q}$-sub-vector-space of $\Bbb{R}$.

Define the set $$\log(\Bbb{P}):=\{\log(p) : p\in\Bbb{P}\}$$ and the $\Bbb{Q}$-vector-space generated by $\log(\Bbb{P})$ $$V_{\Bbb{P}}:=\text{span}_\Bbb{Q}(\log(\Bbb{P}))$$.

Using the unique prime factorization of positive rational numbers in reduced form, one can show that $\log(\Bbb{P})$ is linearly independent. So, it is a basis of $V_{\Bbb{P}}$.

Now consider the function $$f:\log(\Bbb{P})\to\Bbb{Q}\\x\mapsto e^{-x}$$ which maps $\forall p\in\Bbb{P}:f(\log(p))=\frac{1}{p}$.

Since $f$ is defined on a basis of the $\Bbb{Q}$-vector-space $V_{\Bbb{P}}$ and maps into the $\Bbb{Q}$-vector-space $\Bbb{Q}$, there exists a unique functional $h\in V_{\Bbb{P}}^*$ such that $$ \forall v\in\log(\Bbb{P}):h(v)=f(v)$$.

Introduce the set $$\log(\Bbb{N}):=\text{span}_\Bbb{N}(\log(\Bbb{P}))$$ which is the set of linear combinations of basis vectors with natural number coefficients. Then it is relatively easy to see that $$h|_{\log(\Bbb{N})} = \text{LD}\circ\exp|_{\log(\Bbb{N})}$$ where $\text{LD}$ is the logarithmic arithmetic derivative defined on $\Bbb{N}$.

So $h$ is a generalization of $LD$ to a larger space which seems to me to agree with common generalizations of $LD$ to rational or real numbers.

My question now is, what should I study to explore this idea any further? Is there already existing related work to this? I haven't seen anything like this in the papers on the arithmetic derivative that I have read. Have I missed any?

Also, what general theory should I study to better understand this? Obviously I know some basic linear algebra and basic number theory. I also know some basic module theory and basic functional analysis. But pretty much nothing I know tells me where I could go further than this. No idea if $V_{\Bbb{P}}$ could be turned into a Banach space, so I can't really apply Functional Analysis to the problem yet. What I know about functionals is otherwise basically that they form the dual space, but I also don't see here how that would help me. What I know about module theory is also pretty limited to finite-dimensional modules. It also seems to me that it would be pretty helpful to know something about "vector spaces over a monoid" which as far as I know isn't really a thing.

So, any help, direction, whatever is much appreciated, thank you.