That is useful.This actually is a fairly subtle point that usually doesn't matter, but seems to be a source of confusion here. If my answer adds more confusion feel free to ignore it.
Division by an arbitrary voltage is the same thing as selecting your units for measuring the potential difference. For example, suppose we are measuring the height of a man. We might use a meter stick and find that the man is 2 m tall. What this actually means is that the dimensionless ratio between the height of the man and the length of the meter stick is 2. I.e. L(man)/L(meter) = 2 -> L(man) = 2 m. In this sense all dimensionful measurements are actually dimensionless ratios to some standard with the same dimensions.
So if you divide by one Volt you are simply stating that you are using base SI units. On the other hand, if you divide by one statvolt then you are using base CGS units. If you divide by some other voltage then you are using some other units.
In this case the equation works out such that the choice of units does not matter, but that is not always the case. In other situations, where the choice of units matters there will be an explicit division by a non-arbitrary voltage and the argument to the transcendental function will be explicitly non-dimensional. The voltage on the bottom of such an equation defines a kind of "natural" unit of potential difference for the system in question.