The linear transformation of the ability scale that changes the zero is not the only algebraic transformation that has been applied to define the ability scale in the Rasch model. Another is a composite transformation in which first a logarithm of the parameters is taken and all parameter values are subsequently multiplied by a constant. This transformation results in a zero that now seems to be absolute and a unit that is arbitrary. But how is this possible? How can we ever have a class of transformations that leave the fit of the model intact, with one member showing that the scale unit is absolute and another that it is not the unit but the zero that is absolute. Is this not an outright paradox?