Let $k$ a field and $F$ a finite extension of $k(x)$. Let the rational 1-forms $$Fdx = \{ f dx, f \in F\}=\{ f dg, f,g \in F\} $$
(obeying to the rules of $F$-modules, of $k$-linearity and $d1=0$, $d(gh) = gdh+hdg$, plus the algebraic ones : for example with $F = k(x)[y]/(y^2-x^3-x)$ then the algebraic rule is $0 = d(y^2-x^3-x) = (-3x^2-1)dx + 2ydy$ so that $\frac{dy}{dx} = \frac{3x^2+1}{2y} \in F$ and $FdF = Fdx+F dy = Fdx$)
Let $$\ell : F^* \to F dx, \qquad \ell(u) = \frac{du}{u}$$ It is an homomorphism with kernel $K^*$ and for any non-constant $g \in F^*$ then $u \mapsto \frac{\ell(u)}{dg}$ is a logarithm $F^\times/K^\times \to F$ (where $K = \overline{k} \cap F$)
I don't think I can interpret those things in term of Zariski topology.
Is there a notion of continuity or algebraicity that fits to $\ell $ ? Same question with the map to divisors $\text{div} : F^* \to \text{Div}(F)$ and $Fdx \to \text{Div}(F)$. How should I think to those kind of maps in the context of algebraic varieties ?