The distributions of a wide variety of physical, biological, and man-made phenomena approximately follow a power law over a wide range of magnitudes: these include the sizes of craters on the moon and of solar flares, Few empirical distributions fit a power law for all their values, but rather follow a power law in the tail.
Acoustic attenuation follows frequency power-laws within wide frequency bands for many complex media.
Allometric scaling laws for relationships between biological variables are among the best known power-law functions in nature.
One attribute of power laws is their scale invariance. Thus, it follows that all power laws with a particular scaling exponent are equivalent up to constant factors, since each is simply a scaled version of the others.
The demonstration of a power-law relation in some data can point to specific kinds of mechanisms that might underlie the natural phenomenon in question, and can indicate a deep connection with other, seemingly unrelated systems; see also universality above.