I'm not sure that the view that "mathematicians reason syntactically and not semantically" is that unpopular. After all, a lot of modern mathematics nowadays is concerned with how structures are made up of interactions between elements and not the "nature" of the elements themselves.
In that sense, the fact that log(a) + log(b) = log(ab) can be viewed as a homomorphism from an multiplicative group to an additive one (whereas the exponential function is the inverse map).
But I also think it can be illuminating to see that this is not the only "definition" of logarithms, and that there are equivalent definitions. That's precisely the beauty of mathematics, that you can define a number of things and then show them to be exactly the same. There are a number of different ways to define e.g. the exponential function and each of them highlights a different aspect and is interesting to mathematicians working in different disciplines (e.g. the exponential function is also the unique solution to the IVP y'=y with y(0)=1). I don't know if it's possible to teach something like that to children, but it does seem like we're not even trying right now.
In that sense, the fact that log(a) + log(b) = log(ab) can be viewed as a homomorphism from an multiplicative group to an additive one (whereas the exponential function is the inverse map).
But I also think it can be illuminating to see that this is not the only "definition" of logarithms, and that there are equivalent definitions. That's precisely the beauty of mathematics, that you can define a number of things and then show them to be exactly the same. There are a number of different ways to define e.g. the exponential function and each of them highlights a different aspect and is interesting to mathematicians working in different disciplines (e.g. the exponential function is also the unique solution to the IVP y'=y with y(0)=1). I don't know if it's possible to teach something like that to children, but it does seem like we're not even trying right now.