Definition:Completely Additive Function

Let $$K$$ be a field.

Let $$f: K \to K$$ be a function on $$K$$.

Then $$f$$ is described as completely additive iff:


 * $$\forall m, n \in K: f \left({m n}\right) = f \left({m}\right) + f \left({n}\right)$$

That is, a completely additive function is one where the value of a product of two numbers equals the sum of the value of each one individually.

It follows trivially that a completely additive function is also an additive function (in the number theoretical sense), but not necessarily the other way about.

Example
The logarithm is the classic example: from Sum of Logarithms we have that $$\log x + \log y = \log \left({x y}\right)$$.