Fitting data to a power law equation

 This is a common task for experimentalists looking for relationship among data sets and groups of these.

The common feature of data behaving according to a power law equation,

$y=\alpha\,x^\beta$        Eq. (01) 

where $\alpha$ and $\beta$ are free parameters or parameters for adjustment is that once $\ln y$ vs $\ln x$ is plotted, a straight line is obtained.

Schematic representation of data behaving as power law Eq. (01) (left) and data producing a straight line in log log plot (right).

For short, rather than having to use logarithms to get data from a log-log plot, $\alpha$ and $\beta$ can be determined so that all data can be easily represented by Eq. (01).

How to determine the free parameters

Vey easy. Since in the log-log plot the curve becomes a straight line, it should have an equation like,

$\ln y=m\, \ln x + b$        Eq. (02)

where $m$ is the slope and $b$ is the ordinate at which the line cuts the axis. On the other hand, if operate with $\ln$ on Eq. (01) we obtain,

$\ln y=\ln\left| \alpha \, x^\beta \right|$

$\ln y = \ln \alpha + \beta\ln x$        Eq. (03)

Next, comparison of Eqs. (02-03) leads to,

$m=\beta$ and $b=\alpha$        Eq. (04)

Well, this is not all because the real question is not being answered, how can we estimate $\alpha$ and $\beta$? Equation (4) is just a partial answered. A reliable way of finding $m$ and $b$ is by a linear regression using the minimum squares method so that the reader is then referred to,

Minimum squares method

I hope you can find this useful.