What is the significance of using the term "discrete" in discrete logarithm?

I'm trying to clear up my confusion in using the term "discrete" in discrete logarithm. I'm focusing on why the word "discrete" is used to differentiate it from a logarithm.

Wikipedia defines a discrete logarithm as follows:

in any group $G$, powers $b^k$ can be defined for all integers $k$,

and the discrete logarithm $log_b a$ is an integer $k$ such that $b^k = a$.

Is the term discrete added simply to reflect the fact that $k$ in $log_ba=k$ is confined to integers? Or is it a combination of the discrete log being an integer as well as the powers fulfilling the properties of a group $G$?