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.