Power law and log-log graph

A power law has the form

$y = a\cdot x^{n}$

and describes situations in which $y$ scales as a power of $x$. They are everywhere in physics: the force of gravity $F \propto 1/r^{2}$ (exponent $n = -2$), the period of a pendulum $T \propto \sqrt{L}$, that is $L^{1/2}$ (exponent $n = 1/2$), the braking distance $d \propto v^{2}$ (exponent $n = 2$). The exponent $n$ is the law’s signature: stating it precisely means having understood how one quantity depends on the other.

Linearisation: why the log-log graph works

The log-log graph has both axes on a logarithmic scale. Taking the $\log_{10}$ of both sides, the power becomes a multiplication:

$\underbrace{\log_{10}(y)}_{Y} = \underbrace{\log_{10}(a)}_{q} + n\cdot\underbrace{\log_{10}(x)}_{X}$

In the log-log plane $(X,\,Y)$ this is a straight line $Y = q + nX$. The crucial difference from the semi-log case: here the slope is the exponent $n$ of the power law. It is enough to measure the line’s incline to read off the exponent, without needing to know anything about the absolute values.

Recovering the parameters from a log-log graph

$n = \frac{\Delta\log_{10}(y)}{\Delta\log_{10}(x)} \quad(\text{pendenza} = \text{esponente}) \qquad a = 10^{\,q} \ \text{ con } q = \log_{10}(y)\big|_{x=1}$

Take care not to confuse the two tools: on a semi-log graph, a straight line signals an exponential $y = a\,k^{x}$ (the variable is in the exponent); on a log-log graph, a straight line signals a power law $y = a\,x^{n}$ (the variable is in the base). These are different diagnoses for different laws. A concrete application is measuring how the total bounce time depends on the release height, where the log-log graph reveals an exponent $n \approx 1/2$.

Links

Topics: Experimental method Skills: Use of logarithmic scales · Data linearisation · Reading graphs

Related exercises: Worked exercise — Bouncing ball (semilog) · Problem — Bacteria and semilog graph · Problem — Fermi estimate, measurements of g