If the argument of the logarithm is real, then it is positive.

It is defined when the arguments of the logarithm and the square root are not non-positive real numbers.

The argument of the logarithm must be dimensionless, otherwise it is improper, so that the differential entropy as given above will be improper.

In the above "ideal" development, there is a critical point, not at absolute zero, at which the argument of the logarithm becomes unity, and the entropy becomes zero.

This is illustrated in the diagram, where the two black oriented circles are labelled with the corresponding value of the argument of the logarithm used in "z" and (3 "z").

The above equation is a good approximation only when the argument of the logarithm is much larger than unity - the concept of an ideal gas breaks down at low values of V/N.

This reduces the problem to one where the argument of the logarithm is in a restricted range, the interval [1,2), simplifying the second step of computing the fractional part (the mantissa of the logarithm).