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.