It's the automatic value of the option PlotRangePadding, not the PlotRange option, that's going wrong in the ListLogPlot. All that is needed is a user-specified value for the plot range padding on the left-hand side of the plot.

x = Range[0.001, 1000]; y = 5 - Log[x]; ListLogLinearPlot[Transpose[{x, y}], PlotRange -> All, PlotRangePadding -> {{3., Automatic}, Automatic}] 

It's the automatic value of the option PlotRangePadding, not the PlotRange option, that's causing the problem in the ListLogPlot. The default Automatic value adds 4% to end-point values of the range. This is nowhere enough for the ListLogPlot with its extreme range. 

All that is needed are user-specified values for the plot range padding on the right- and left-hand sides of the plot.

x = Range[0.001, 1000]; y = 5 - Log[x]; ListLogLinearPlot[Transpose[{x, y}], PlotRange -> All, PlotRangePadding -> {{10., .1}, Automatic}]