1) entropy is a mathematical concept. Physics apply it, like any other mathematical one
2) math entropy does not change, everything is reversible, as it is based on known quantities
3) entropy measures how far we are from perfectly knowing a system
4) logarithm is chosen to measure entropy because of its convenient properties, not because of any deeper law. It could be a sum, a product etc
5) the Second thermodynamic law is a tautology: "every system tends to higher entropy state because it is more common" becomes "the more common is the more common"
5) there is no every system, there is only the universe in which all systems increase in entropy even if individual objects (block of ice) do not. You're bleeding off entropy into the universe you can never get back.
i think those convenient properties are the sign of some deeper law. you do want to work with the exponential and logarithm, same as in so many other places. you don’t necessarily want some other generalized “entropy”.
Thermodynamics is a young and poorly thought out science.
My beef is the adoption of bad names and pseudo-philosophy. In Gibbs free energy equation, there is a term called entropy but it is nothing more than the value for heat transferred for the given elements and given constant temperature and derived from experimental observation. Instead of calling it entropy, they could have called it Clausius constant. No need to confuse generations of students.
1) entropy is a mathematical concept. Physics apply it, like any other mathematical one
2) math entropy does not change, everything is reversible, as it is based on known quantities
3) entropy measures how far we are from perfectly knowing a system
4) logarithm is chosen to measure entropy because of its convenient properties, not because of any deeper law. It could be a sum, a product etc
5) the Second thermodynamic law is a tautology: "every system tends to higher entropy state because it is more common" becomes "the more common is the more common"