# Power law

A power law relationship between two scalar quantities x and y is any such that the relationship can be written as

[itex]y = ax^k\,\![itex]

where a (the constant of proportionality) and k (the exponent of the power law) are constants.

Power laws can be seen as a straight line on a log-log graph since, taking logs of both sides, the above equation is equal to

[itex]\log(y) = k\log(x) + \log(a)\,\![itex]

which has the same form as the equation for a line

[itex]y = mx+c\,\![itex]

Power laws are observed in many fields, including physics, biology, geography, sociology and economics. Power laws are among the most frequent scaling laws that describe the scaling invariance found in many natural phenomena.

Examples of power law relationships:

Examples of power law probability distributions:

These appear to fit such disparate phenomena as the popularity of websites, the wealth of individuals, the popularity of given names, and the frequency of words in documents.

• Art and Cultures
• Countries of the World (http://www.academickids.com/encyclopedia/index.php/Countries)
• Space and Astronomy