ListLogLinearPlot for the whole real numbers

Question

I have the followin data

llvaluefull = {{-10000, 0.9955201558795129`}, {-5000, 
0.9918661998405411`}, {-1000, 0.9689994447758001`}, {-750, 
0.960994816269547`}, {-500, 0.9464348207676564`}, {-400, 
0.9364572120784225`}, {-300, 0.9211596225765791`}, {-200, 
0.8942080072544306`}, {-100, 0.8309553835209477`}, {-75, 
0.797892513207336`}, {-50, 0.7452046521006773`}, {-40, 
0.713946015544555`}, {-30, 0.6727095338844533`}, {-20, 
0.6169570669713804`}, {-15, 0.5830218293040762`}, {-10, 
0.5491452142752082`}, {-5, 0.5330050345080709`}, {-3.5, 
0.5370096909263407`}, {-2.5, 0.5425758605796321`}, {-1.5, 
0.5501718631932083`}, {-2/3, 0.5576614234516881`}, {-0.25, 
0.5616729002916181`}, {-0.125, 0.5629013853879679`}, {-0.0125, 
0.5640017268531379`}, {-10^-2, 0.564051437097455`}, {-10^-3, 
0.56409191181635`}, {-10^-4, 0.5639969267400813`}, {-10^-5, 
0.5639866088126503`}, {10^-5, 0.5640044975410387`}, {10^-4, 
0.5639945512825674`}, {0.001, 0.5642321885539248`}, {0.01, 
0.564252028156347`}, {0.125, 0.5666028574308888`}, {0.25, 
0.5654228175723596`}, {2/3, 0.5708558542568287`}, {1.5, 
0.5793912853563281`}, {2.5, 0.5896754113240948`}, {3.5, 
0.5999259016520363`}, {4, 0.6050389881798164`}, {10, 
0.6644240081298314`}, {15, 0.7062425236595297`}, {20, 
0.7394977911965404`}, {30, 0.7870306109737224`}, {40, 
0.8188909726186043`}, {50, 0.8417288556450642`}, {75, 
0.8781587302178719`}, {100, 0.8998929170621898`}, {250, 
0.9488186518137977`}, {500, 0.9701978007543405`}, {1000, 
0.9829942727951643`}, {2500, 0.9920867688907514`}, {5000, 
0.9956233626141124`}};

I am able to plot this on the real line including negative numbers with

ListLinePlot[llvaluefull]

But I need x-axis to be logarithmic. Therefore I use

ListLogLinearPlot[llvaluefull, Joined -> True]

However, this gives me a figure only for the positive real numbers. Is there a way to have a logarithmic scale for the both positive and negative real numbers on the same figure?

By the way, I know that logarithm is defined on the positive numbers. I only need log scale in both directions.

For the only positive part (by deleting the negative ones in the list) I get

Answer 1

According to the comment below your question I believe this does what you want:

scale    = Sign[#] Log[1 + Abs@#] &;
invscale = Sign[#] (Exp[Abs@#] - 1) &;

ListLinePlot[
 llvaluefull,
 ScalingFunctions -> {{scale, invscale}, Identity}
]

ScalingFunctions works in ListLinePlot in Mathematica 10.0.2, but it is not officially supported. It may not work in earlier versions.

---

Adapting the code I used for How do I make a log plot where the plot is logarithmic in the distance from the X-Axis (including negative values)? you can arbitrarily zoom the zero region with:

logify[_][x_ /; x == 0] := 0
logify[off_][x_] := Sign[x] Max[0, (off + Re@Log@x)/off]

inverse[off_][x_] := Sign[x] Exp[(Abs[x] - 1) off]

logscale[n_] := {logify[n], inverse[n]}

Examples:

ListLinePlot[llvaluefull, ScalingFunctions -> {logscale[1], Identity}]

With logscale[7]:

With logscale[22]:

Answer 2

One way is through a bilogarithmic plot.

Define

    bilog[val_, cut_: 1., ff_: .25] := Module[
   {out},
   out = If[Abs[val] <= cut,
     ff val,
     Sign[val] Log10[Abs[val]]
     ]
   ];

for the data and

 blvs[{rl_, rh_}, cut_: 1] := Module[
  {out, lin, lgn, lgp, lgt, lgm, lgo, tik, tkn, tkp},
  lin = Range[-.9 cut, .9 cut, cut/10];
  lgp = Range[Log10[cut], Log10[rh], 1];
  lgn = Range[Sign[rl] Log10[Abs@rl], Log10[cut], 1];
  lgm = {#, Sign[#] 10^(Abs@#)} & /@ Join[lgn, lgp];
  lgo = Log10[Range[1, 9, 1.]];
  tkn = Sort@-(lgo + # & /@ Sort@Abs@lgn);
  tkp = Sort@+(lgo + # & /@ Sort@Abs@lgp);
  tkn = {#, ""} & /@ Flatten@tkn;
  tkp = {#, ""} & /@ Flatten@tkp;
  tik = Join[tkn, tkp, lgm]
  ]

for the frame ticks, then

blv = {N@bilog[#[[1]], 1.], #[[2]]} & /@ llvaluefull;

and

ListPlot[blv, Frame->True,
 FrameTicks -> {{Automatic, None}, {blvs[{-10000, 10000}, 1], None}}]

gives you the plot I think you're after. Sans labels.

You'll also want to specify GridLines to make it clear that the region < Abs@cut has a linear scale.

The ff variable in the definition of bilog is cearly a fudge factor to scale the linear section so that it looks right.

Obviously, there are better ways of doing this job, and they are likely to involve the superposition of three properly-sized graphs for the appropriate regions. An exercise for the reader perhaps.

Another popular (at least in the geophysics community) transformation is through ArcSinh, but I'll leave definition of frame ticks as another exercise for the reader.