Consider the functional equation (for $f: \mathbb{C} - 0 \rightarrow \mathbb{C}, (c,x) \in \mathbb{C}^2 $)
$$ f(cx) = f(x) + 1 $$
The "standard solution to this is the logarithm $\log_c(x)$ (although there's an infinite family of additional solutions that we wont concern ourselves with right away).
I now decided to consider the following two parameter
$$ f(ax + b) = f(x) + 1 $$
I define $\text{glog}_{a,b}(x)$ as the solution to it ("Generalized-Logarithm), i.e.
$$ \text{glog}_{a,b} (ax + b) = \text{glog}(x) + 1 $$
I feel pretty certain someone has defined this function before me, and studied its properties (maybe even defined it as a taylor series for suitable choice of a,b). Does anyone know what it might be called in the standard literature/where to find it?