# $\log_q \log_q$ plot instead of $\log \log$ plot

By defining the $\log_q$ as: $$\log_q(x)=\frac{x^{1-q}-1}{1-q}, x \geq 0, q \neq 1$$ how would it be possible to have the equivalent of the LogLogPlot command, meaning that the axes are now defined as $\log_q (x)$ and $\log _q f(x)$?

The reason I am asking this, is because I would like to make a plot of the Tsallis q-Gaussian distributions which are built-in Mathematica Tsallis Q Gaussian and then the equivalent of the LogLogPlot in order to show that these distributions about power laws on a $q$-logarithmic scale.

Thank you.

• Why not use ParametricPlot[]? Jul 11 '16 at 14:47
• @J.M. Can you please explain in more detailed what do you mean by this suggestion? As far as I am concerned I just know that LogLogPlot defines logarithmic scales on both axes but I am not familiar with the way Mathematica does that, therefore my problem is my lack of ability to manipulate this command. Jul 11 '16 at 14:54
• A hint: compare LogLogPlot[Exp[x], {x, 1, 5}] and ParametricPlot[{Log[x], Log[Exp[x]]}, {x, 1, 5}, AspectRatio -> 1/GoldenRatio]. (Worry about the ticks later; concentrate on what these transformations are doing mathematically.) Jul 11 '16 at 14:58
• there is nothing special about LogLog plot. You can always make transformation and then produce special ticks/gridlines, something like this Jul 11 '16 at 15:49
• Calling that function $\log_q$ is really misleading. That notation is standard for the base-$q$ logarithm. Jul 11 '16 at 22:09

You might be able to make use of ScalingFunctions.

lq[q_ /; q != 1][x_] := (x^(1 - q) - 1)/(1 - q)
invlq[q_][z_] := Abs[(1 + z - q z)^(1/(1 - q))]

spec[q_] := {{lq[q], invlq[q]}, {lq[q], invlq[q]}};

Plot[x^2, {x, 1, 30}, PlotRange -> All, ScalingFunctions -> spec[0.8]]


This works in Mathematica 10.1 but it is not officially supported, as indicated by the Option name being highlighted in red.

• In this case, the OP seems to want to plot tqpdf[x_, q_] = PDF[TsallisQGaussianDistribution[q], x]; maybe you can use that example instead (making sure you are using the same q for it and lq[]). For reference: here are complete definitions you can use for spec[]. Jul 11 '16 at 15:11
• @J.M. I really don't know how that should look so I feel uncomfortable with it. Jul 11 '16 at 15:15
• @Mr.Wizard Thank you, I will take some time to play around with it, perhaps also use the TsallisQGaussianDistribution to see if I get what is expected by it and then I will get back to you. Jul 11 '16 at 15:40