Intuitive understanding of logarithms

I know logarithms are supposed to be the inverse of exponential functions, and while this makes sense, it seems to me that a more intuitive and significant property is

$\mathrm{log}(ab)=\mathrm{log}(a)+\mathrm{log}(b)$

So in this way, the logarithm is a fundamental relationship between addition and multiplication. Should logarithms in schools be taught this way? Should I think of them primarily in this way?

EDIT: This probably related to the fact that the only continuous functions $f$ that satisfy $f(x+y)=f(x)f(y)$ are exponential functions (there are apparently some super-weird non-continuous non-exponential functions that satisfy that multiplicativity but I have no idea what they are).