The function y=x^{p} is "scale-invariant" in the following sense.
Consider an interval such as (x,2x), where y changes from x^{p}
to 2^{p}x^{p}. Now scale x by a scale factor
a. We look at the interval (ax,2ax) where y changes from (ax)^{p}
to 2^{p}(ax)^{p}. We see that y in this interval
is the same (except for a scale change of a^{p}) 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=Ax^{2} + Bx can be
written as A(x+B/2A)^{2} -B^{2}/4A. If the new variable
X=x+B/2A is scaled to aX, then y(aX)=a^{2}Y(X)+(a^{2}-1)(B^{2}/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/T_{c}). 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

lim_{p-->0} (x^{p}-1)/p = log x

which shows that logs are a special case of power law functions with
power 0.