DensityPlot with logarithmic xy-axes and logarithmic color scale

Question

I would like to do a DensityPlot with logarithmic xy-axes and logarithmic color scale for the function

\begin{align} S(\epsilon_{v},\epsilon_{\phi}) = \frac{\pi}{\epsilon_{v}} \frac{\sinh \left(\frac{2 \pi \epsilon_{v}}{\pi^{2} \epsilon_{\phi}/6} \right)}{\cosh \left(\frac{2 \pi \epsilon_{v}}{\pi^{2} \epsilon_{\phi}/6} \right) - \cos \left(2 \pi \sqrt{\frac{1}{\pi^{2} \epsilon_{\phi}/6} - \left( \frac{\epsilon_{v}}{\pi^{2} \epsilon_{\phi}/6} \right)^{2}} \right)}\\ ~\\ \epsilon_{v} \to x , \epsilon_{\phi} \to y~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \end{align}

[Background: Considering velocity dependent Dark Matter interaction this function (Sommerfeld enhancement) arises when solving the corresponding Schrödinger equation with a Hulthén potential]

While searching for a solution I found these two posts:

1) Logarithmic scale in a DensityPlot and its legend

2) DensityPlot with one log axis

But I am searching in some sense a combination of these two ideas.

Alas I started using Mathematica only a few weeks ago and I am not familiar with all its features. Hence, I came to no satisfactorily result yet and would be very happy if anyone can help.

Here the Matematica code which I in principle took from 1):

f[x_, y_] := 
  Pi/x * Sinh[
     12*x/(Pi*y)]/(Cosh[12*x/(Pi*y)] - 
      Cos[2*Pi*Sqrt[ 6/(Pi*Pi*y) - (6*x/(Pi*Pi*y))^2]]);



plotter[min_, max_, NumberOfTicks_] := 
 DensityPlot[f[x, y], {x, 10^(-6), 10^(0)}, {y, 1/(10^4), 1/(10^3)}, 
  PlotPoints -> 10, PlotRange -> Full, ColorFunctionScaling -> False, 
  ColorFunction -> (ColorData["Rainbow"][
      LogarithmicScaling[#, min, max]] &), 
  PlotLegends -> 
   BarLegend[{ColorData["Rainbow"], {0, 1}}, LegendMarkerSize -> 340, 
    Ticks -> ({LogarithmicScaling[#, min, max], 
         ScientificForm[#, 2]} & /@ (min (max/min)^
          Range[0, 1, 1/NumberOfTicks]))]]
plotter[1, 10^(6), 6]

Answer 1

wxffles suggestion to use a helper function to do the coordinate transformation seems quite effective:

f[x_, y_] := Pi/x*Sinh[12*x/(Pi*y)]/(Cosh[12*x/(Pi*y)] - 
      Cos[2*Pi*Sqrt[6/(Pi*Pi*y) - (6*x/(Pi*Pi*y))^2]]);

helper[x_, y_] = f[10^x, 10^y];

LogarithmicScaling[x_, min_, max_] := Log[x/min]/Log[max/min]
plotter[min_, max_, NumberOfTicks_] := DensityPlot[

  (* Modification to use the helper function *)
  helper[x, y],
  {x, -6, 0}, {y, -4, -3},
  (* end modification *)

  PlotRange -> Full,
  ColorFunctionScaling -> False,
  ColorFunction -> (ColorData["Rainbow"][
      LogarithmicScaling[#, min, max]] &),

  PlotLegends ->
   BarLegend[{ColorData["Rainbow"], {0, 1}}, LegendMarkerSize -> 340, 
    Ticks -> ({LogarithmicScaling[#, min, max], 
         ScientificForm[#, 2]} & /@ (min (max/min)^
          Range[0, 1, 1/NumberOfTicks]))]
  ]

With these in place:

plotter[1, 1*^6, 6]

Answer 2

This uses the CustomTicks package available here, and the nonLinearDensityPlot function defined here. See below for a version that does not use these add-ons

loglogDensityPlot[func_, {x_, xmin_, xmax_}, {y_, ymin_, ymax_}, 
    plotopts : OptionsPattern[{DensityPlot, nonLinearDensityPlot}]] :=
  nonLinearDensityPlot[
  func /. {x -> Exp[x], y -> Exp[y]}, {x, Log@xmin, Log@xmax}, {y, 
   Log@ymin, Log@ymax},
  FrameTicks -> {Charting`ScaledTicks[{Log, Exp}],
    Charting`ScaledTicks[{Log, Exp}]},
  ImagePadding -> {{Automatic, None}, {Automatic, None}},
  Evaluated -> True,
  plotopts]

By default this function uses an ArcSinh scaling function, that doesn't look great for your data

loglogDensityPlot[f[x, y], {x, 1/1000000, 1}, {y, 1/10000, 1/1000}]

To change this, you need to provide a scaling function that maps the values your function can take to the range {0,1}. The arguments of the scaling function are {value, scalefactor, maxval, minval}. So to get linear scaling we use ((#1 - #4)/(#3 - #4) &). For the log scaling we use

loglogDensityPlot[
 f[x, y], {x, 10^(-6), 10^(0)}, {y, 1/(10^4), 1/(10^3)}, 
 "ScalingFunction" -> (Log[#1/#4]/Log[#3/#4] &)]

Here is a version that doesn't use external packages, but I can't get a nice legend out of it. It is similar to MarcoB's answer,

loglogDensityPlot[func_, {x_, xmin_, xmax_}, {y_, ymin_, ymax_}, 
  opts : OptionsPattern[]] := 
 DensityPlot[Log@(func /. {x -> Exp[x], y -> Exp[y]}),
  {x, Log@xmin, Log@xmax}, {y, Log@ymin, Log@ymax},
  FrameTicks -> {Charting`ScaledTicks[{Log, Exp}],
    Charting`ScaledTicks[{Log, Exp}]},
  opts, PlotRange -> All,
  ImagePadding -> {{Automatic, None}, {Automatic, None}},
  Evaluated -> True
  ]

loglogDensityPlot[f[x, y], {x, 1/1000000, 1}, {y, 1/10000, 1/1000}, 
 PlotPoints -> 100]