You are asserting (not testing) that x cannot actually be zero but rather that you simply cannot measure (or control) it below a certain level.
If it is a dose, for instance, you are asserting that there is always some of the compound in the background environment. If it is light, you are asserting that there is some background level of photons, etc.
The problem is, your $c$ is a guess and may bias the results even if your assertion is true. You may want to try several values.
Your model is related to $Y = e^a (x+c)^b \times \epsilon$.
If you intend to fit $Y = e^a (x+c)^b + \epsilon$ then how you fit the model matters. You maybe could estimate $c$ as a parameter but this could be problematic and may not be worth the bother.