According to Wikipedia: “In algebra, a **homomorphism** is a structure preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces).”

This is often represented by the equation:

$f(x \cdot y) = f(x) \cdot f(y)$

Personally, I prefer to use this equation:

$f(x + y) = f(x) \times f(y)$

In my opinion it more accurately portrays that the operations aren’t necessarily the same. It can be confusing, but this seems to make a clearer distinction between the operations.

As a side note on logarithms.

“**Classically, a logarithm is a partially-defined smooth homomorphism** from a multiplicative group of numbers to an additive group of numbers.”

This is according to nLab. And nLab is, in my opinion, a wonderful resource.