Math Help - Plots & Distributions

1. Plots & Distributions

I seek advise on the following comment:

"Observing the log-log plot we can see that degree distribution fits well to a power law distribution."

The author of the comment proposes that the nature of the distribution only becomes ostensible if seen at log scale.

My question is the following:

What would prompt a conductor of a given study to investigate functions of the X- & Y- axis variables? Do certain specific properties of the data encourage further investigation? If yes, which functions other than log should be considered?

Best,
wirefree

2. Log-log plots

Hello wirefree
Originally Posted by wirefree
I seek advise on the following comment:

"Observing the log-log plot we can see that degree distribution fits well to a power law distribution."

The author of the comment proposes that the nature of the distribution only becomes ostensible if seen at log scale.

My question is the following:

What would prompt a conductor of a given study to investigate functions of the X- & Y- axis variables? Do certain specific properties of the data encourage further investigation?
A very brief answer would be that a log-log plot is useful if you think you may have a function of the form

$f(x) = ax^n$

and you want to find out (a) whether this is so; (b) if so, the values of $a$ and $n$.

If we take logs of both sides:

$\log(f(x)) = \log(ax^n)$

$= \log(a) + n \log(x)$, using the laws of logarithms.

Thus, if the function is of the form $f(x) = ax^n$, the log-log plot will produce a straight line graph, whose gradient gives the value of $n$, and intercept the value of $\log(a)$. (So if the base of the logarithm is 10, and the intercept is $c$, the value of $a$ is given by $a = 10^c$.)

Other applications include the investigation of functions that are themselves logarithmic in nature - the bel and decibel for the measurement of acoustic power, for example.

Here's the Wikipedia article (which perhaps you've already looked at!): Logarithmic scale - Wikipedia, the free encyclopedia.

If yes, which functions other than log should be considered?
I can't think of any others that would be useful!

I hope that helps to shed a little light.