Rectangle fillings obscures the plot

Question

I used two dashed lines in the following codes to separate my plot into three blocks filled with color. But the fillings would obscure the curves (EtaH[n] and EpsilonH[n]).

fill1 = Rectangle[{zerosofeta[[1]], Log@(10^(-12))}, {zerosofeta[[2]],
     Log@4}];
fill2 = Rectangle[{0, Log@(10^(-12))}, {zerosofeta[[1]], Log@4}];
fill3 = Rectangle[{zerosofeta[[2]], Log@(10^(-12))}, {70, Log@4}];

LogPlot[{\[Epsilon]H[n],Abs[\[Eta]H[n]]}, {n, 0, 70},PlotRange -> {10^(-11), 4}, 
 Epilog -> {{Directive[Black, 
      Dashed], 
     Line[{{zerosofeta[[1]], Log@(10^(-12))}, {zerosofeta[[1]], 
        Log@4}}]}, {Directive[Black, Dashed], 
     Line[{{zerosofeta[[2]], Log@(10^(-12))}, {zerosofeta[[2]], 
        Log@4}}]},{LightGreen, fill1}, {LightBlue, fill2}, {LightBlue, fill3}}]

where zerosofeta is a two-element list used to put the roots of EtaH[n].

I have tried another way

curve = LogPlot[{\[Epsilon]H[n],Abs[\[Eta]H[n]]}, {n, 0, 70}, 
   PlotRange -> {10^(-11), 4}];
rectangles = 
  Graphics[{{LightGreen, fill1}, {LightBlue, fill2}, {LightBlue, 
     fill3},{Directive[Black, 
      Dashed], 
     Line[{{zerosofeta[[1]], Log@(10^(-12))}, {zerosofeta[[1]], 
        Log@4}}]}, {Directive[Black, Dashed], 
     Line[{{zerosofeta[[2]], Log@(10^(-12))}, {zerosofeta[[2]], 
        Log@4}}]}}];
Show[rectangles, curve]

but the axes disappear.

Do you have any idea?

updated: The functions etaH[n] and epsilonH[n] are composed of InterploatingFunction which is the numerical solution of a PDE. But I might have a generic one to replace etaH[n] for error finding

eta[n_] := Module[{e0 = 0.02, ni0 = 0.0, no0 = 33.2, s0 = 1.,
e1 = -6.3, ni1 = 33.2, no1 = 35.7, s1 = 0.5,
e2 = 0.3, ni2 = 35.7, no2 = 55, s2 = 1., ef = 3, nif = 55, nof = 65, sf = 2},
(e0/2) * (Tanh[(n - ni0)/s0] - Tanh[(n - no0)/s0]) +
(e1/2) * (Tanh[(n - ni1)/s1] - Tanh[(n - no1)/s1]) +
(e2/2) * (Tanh[(n - ni2)/s2] - Tanh[(n - no2)/s2]) +
(ef/2) * (Tanh[(n - nif)/sf] - Tanh[(n - nof)/sf])]

Answer 1

Epilog is something that will bring the argument at the front of the LogPlot layer. If you put your rectangles in the Epilog then they will mask your functions.

You can instead use Prolog to have them in the background. This avoids the need to use multiple plots in the solution given by @bmf (although you should keep that method in mind as it is a very good one as well)

zerosofeta = {30, 
   38}; (*i have chosen the points to match yours roughly*)

fill1 = Rectangle[{zerosofeta[[1]], Log@(10^(-12))}, {zerosofeta[[2]],
     Log@4}];
fill2 = Rectangle[{0, Log@(10^(-12))}, {zerosofeta[[1]], Log@4}];
fill3 = Rectangle[{zerosofeta[[2]], Log@(10^(-12))}, {70, Log@4}];

LogPlot[{Sin[n]^2, 0.01 Sqrt[Tanh[n]]}, {n, 0, 70}, 
 PlotStyle -> {Blue, Red}, PlotRange -> {1. 10^-12, 4}, 
 PlotLegends -> {"func_1", "func_2"},
 
 (*create the prolog background*)

 Prolog -> {{LightGreen, fill1}, {LightBlue, fill2}, {LightBlue, 
    fill3}},
 
 (*now add your epilog with the lines-you could instead use Gridlines*)

 Epilog -> {{Black, Dashed, 
    Line[{{zerosofeta[[1]], Log[1. 10^-12]}, {zerosofeta[[1]], 
       Log[4]}}]},
   {Black, Dashed, 
    Line[{{zerosofeta[[2]], Log[1. 10^-12]}, {zerosofeta[[2]], 
       Log[4]}}]},
   Inset["Phase I", {15, Log[10^-7]}],
   Inset["Phase II", {33.8, Log[10^-7]}],
   Inset["Phase III", {55, Log[10^-7]}]
   }]

Note that you could have also used Gridlines instead of Lines and an Epilog.

Lines are generally not the best to use with LogPlot as there is a small (but a bit annoying ;) ) scaling that you need to perform. You can find this SE link useful.

Answer 2

If I understand the problem correctly you want something like this:

So, essentially you want to use Overlay.

The code:

zerosofeta = {-2, 10};
f[n_] := Log[1/Sqrt[n - 1]]^3
g[n_] := 1/Log[n + 2]^2
fill1 = Rectangle[{zerosofeta[[1]], Log@(10^(-12))}, {zerosofeta[[2]],
     Log@4}];
fill2 = Rectangle[{0, Log@(10^(-12))}, {zerosofeta[[1]], Log@4}];
fill3 = Rectangle[{zerosofeta[[2]], Log@(10^(-12))}, {70, Log@4}];
p1 = LogPlot[{f[n], Abs[g[n]]}, {n, 0, 70}, 
   PlotRange -> {10^(-11), 4}, 
   Epilog -> {{Directive[Black, Dashed], 
      Line[{{zerosofeta[[1]], Log@(10^(-12))}, {zerosofeta[[1]], 
         Log@4}}]}, {Directive[Black, Dashed], 
      Line[{{zerosofeta[[2]], Log@(10^(-12))}, {zerosofeta[[2]], 
         Log@4}}]}}];
p2 = LogPlot[{f[n], Abs[g[n]]}, {n, 0, 70}, 
   PlotRange -> {10^(-11), 4}, 
   Epilog -> {{Directive[Black, Dashed], 
      Line[{{zerosofeta[[1]], Log@(10^(-12))}, {zerosofeta[[1]], 
         Log@4}}]}, {Directive[Black, Dashed], 
      Line[{{zerosofeta[[2]], Log@(10^(-12))}, {zerosofeta[[2]], 
         Log@4}}]}, {LightGreen, fill1}, {LightBlue, 
      fill2}, {LightBlue, fill3}}];
Overlay[{p2, p1}]