The function y=xp is "scale-invariant" in the following sense. Consider an interval such as (x,2x), where y changes from xp to 2pxp. Now scale x by a scale factor a. We look at the interval (ax,2ax) where y changes from (ax)p to 2p(ax)p. We see that y in this interval is the same (except for a scale change of ap) as y in the unscaled interval.
Most functions do not behave this way. Consider y=exp(x), which goes [from exp(x) to exp(2x)] over the interval (x,2x), and [from exp(ax) to exp(2ax)] in the scaled interval (ax,2ax). There is no scale factor that can be removed from y to make the second interval of y appear the same as the first.
The sum of two different powers is usually not scale invariant.
However, special cases may still be. y=Ax2 + Bx can be
written as A(x+B/2A)2 -B2/4A. If the new variable
X=x+B/2A is scaled to aX, then y(aX)=a2Y(X)+(a2-1)(B2/4A).
In other words, the function y(x) is scale invariant (with scaling exponent
2) after finding the right scaling variable X and allowing for a scale-dependent
shift of y. In this sense, the thermodynamic functions seem to display
scale invariance around the critical point (for systems that have a critical
point!), with the scaling variable t=(1-T/Tc). Note that
the logarithm y=log x obeys y(ax)=y(x) + log a. It is scale invariant
with exponent 0 (and a scale-dependent shift.) This is related to
the famous formula
limp-->0 (xp-1)/p = log x
which shows that logs are a special case of power law functions with power 0.