Hey g4143 and welcome to the forums.
You do actually raise indirectly an interesting point about the duality between multiplication and addition, which is seen the logarithmic function where log(xy) = log(x) + log(y) so in this sense, there is actually a bridge to relate multiplication purely to addition.
However with this said, even if you are able to eventually reduce all log(xy) down to summations where x and y are real numbers (or even complex ones), you might want to think about exponentiation: log(x^y) = y*log(x) and although you could do a double log to convert to an addition, this raises an issues of how to deal with this.
I am basically being generous in the above assumption and am assuming you have a way to eventually break any product of real numbers into an addition which you can calculate quantities without resorting to addition, and basically I think this is going to involve a lot of abstract symbolic relations between things to get around the fact that calculating explicitly the logarithm requires a power series that has powers (i.e. repeated multiplication) if you want to do a specific finite computation.
The analogue that I am making can be seen with what people do with the sine and cosine functions: we know for example some exact expressions for sine and cosine (like for various fractions of pi) and we can use all the various sine and cosine expressions to get exact expressions for a particular quantity without needing to actually use the power series and I think if you wanted to do the above, you would need to do the exact same thing but for logarithms.
Interesting enoughly, there is a direct connection with exponentiation and the trig functions, so there is actually a bridge that you can build and see if you can cross.
Problem is though, that we don't really have a way currently that is known of, to just get the exact value for the sine of any number on the real line in exact form, so if you want to do your thing, you need to solve this problem first (I still think it's actually possible to do this, but it's going to be rather difficult).
But this would only look at being able to express arithmetic operations in terms of only two operations and a lot of this would require a completely complex symbolic framework.
The other thing you need to look at, as hinted above by Number Nine, is the abstract stuff that does not necessarily correlate to arithmetic in a direct way like sets and the rest of the other abstract things built on general sets.
Sets have a duality too but it deals with intersections and unions, and sets don't work like numbers: they are a completely different way of looking at things because sets do not have rank or order like numbers do. You can introduce relations and things on various sets but it is by no means required to do so.
So since you have these sets that do not have any notion of rank, or comparison in the arithmetic sense, you can't really apply that kind of thinking to all the stuff based on sets which is pretty much all of mathematics at some level.
I suggest you take a look at the set framework and the two operations of intersections and unions to get a feel of the ground-work for modern abstract mathematics and then consider that in light of your question.