Definition:Completely Additive Function

Definition

Let $\left({R, +, \times}\right)$ be a ring.

Let $f: R \to R$ be a mapping on $R$.

Then $f$ is described as completely additive iff:

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

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, where in this case $\left({R, +, \times}\right)$ is the set of real numbers.

From Sum of Logarithms we have that $\ln x + \ln y = \ln \left({x y}\right)$.