Actually a good question. Logarithm-of-states entropy is also called logical entropy and there's been no end of confusion on these threads (and in my head) about how equivalent that concept is to thermodynamic entropy (useless background energy).
A good starting point is here. The concepts are highly related but a key difference is that logical entropy depends upon the arbitrary concept of "recognizeable states." Because of this you can have a non-zero change in logical entropy accompanied by a zero change in thermodynamic entropy as Klyce shows on that web page.
If one is changing while the other is not, even though they both represent the same thing when it comes down to it, would this not be evidence of a limitation of the method of measurement as applied to logical entropy? How would that impact upon Thorne's formulation?