Are "logarithm types" a thing?
Logarithm types are definitely a thing and have been noticed before by a number of people. In functional programming, a Type -&gt; Type functor has a logarithm if it's representable, and then the logarithm is the representing object. See also this, this, and this. You are correct about the exponential functor being the bag functor, see e.g. this, where it's described as the fixed point of the derivative operation on endofunctors.