You can define any logarithm with the following: $$\log(xy)=\log(x)+\log(y)$$ $$\log(1)=0$$ and then just define $\log(b)=1$ for any base $b$. This looks like a group homomorphism $$\log:\mathbb{R}^{\times}\to \mathbb{R}$$ where $\mathbb{R}^\times$ is $(0, \infty)$ under multiplication, and $\mathbb{R}$ is just the reals under addition, which is operation preserving by definition. What is sort of interesting is that these are "two parts of a field", one being the 'multiplicative part' and and the other being the 'additive part'. Anyways, my question is this:
Can we apply this to any field $F$, as in can we define a group homomorphism $\log : F^\times \to F$, and get something that acts 'logarithmic'? More specifically, is this at all a good approach to generalize logarithms?
I see some problems with this, for example what would this look like for finite fields? Also, $\log$ only maps the positive reals under multiplication to the reals under addition, and lots of fields don't have a good notion of 'positive'. Is there any way to save this?