Why addition/multiplication are not considered a mathematical functions?

Question

Why we always consider exponentiation/logarithm to be a functions of two variables, but the same terminology never applies to addition/multiplication. Is it just because it's never useful or am I fundamentally wrong thinking that addition/multiplication are just basic mathematical functions of two variables?

Answer 1

Addition, multiplication, and exponentiation are "functions". Furthermore, they are all hyperoperations belonging to the hyperoperation sequence, which has no upper limit.
Basically, we can state that addition is to multiplication as multiplication is to exponentiation as exponentiation is to tetration as tetration is to pentation and so forth.

Thus,
Hyper-$0$ $\rightarrow$ zeration;
Hyper-$1$ $\rightarrow$ addition;
Hyper-$2$ $\rightarrow$ multiplication;
Hyper-$3$ $\rightarrow$ exponentiation;
Hyper-$4$ $\rightarrow$ tetration;
$\dots$

Now, any time that you climb one step on the ladder of hyperoperations, you lose something, a property... let us say, addition is commutative, multiplication is commutative, exponentiation is not (e.g., $3^2 \neq 2^3$) and so on for hyper-$k$, where $k>2$, even if I have shown that tetration has a peculiar property (the constancy of its congruence speed for any base which is not a multiple of $10$).
This last observation let us point out that exponentiation, tetration, and so forth are binary operators and the commutative property doesn't hold for them, since it doesn't hold for hyper-$2$.