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 $$\log (ab) = \log(a)+\log(b)$$
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).
The property you posted is derived from the exponent property $$(a^n)(a^m)=a^{n+m}$$ therefore we can see that logarithms and exponents are essentially the same, however in a different notation. I think that makes more sense than them being a relationship between multiplication and addition.
If: $$10^n=A$$ Then: $$log(A)=n$$ And if: $$10^m=B$$ Then: $$logB=m$$
If we take $$10^c=(10^n)(10^m)=10^(m+n)$$ It follows to say that: $$c=n+m$$ therefore: $$log(AB)=logA+logB$$