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}
    ]

![enter image description here][1]

[`ScalingFunctions`](http://reference.wolfram.com/language/ref/ScalingFunctions.html) works in [`ListLinePlot`](http://reference.wolfram.com/language/ref/ListLinePlot.html) in *Mathematica* 10.0.2, but it is not officially supported.  It may not work in earlier versions.


  [1]: https://i.sstatic.net/Pc0zd.png