According to the comment below your answer 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. I may not work in earlier versions.