How can I make a NumberLineLogPlot'?

Use the option Ticks

intervals = {{1 <= x <= 10}, {10 <= x <= 100}, {100 <= x <= 1000}};

logIntervals = intervals /. {n_?NumericQ :> Log10[n]}

(* {{0 <= x <= 1}, {1 <= x <= 2}, {2 <= x <= 3}} *)

NumberLinePlot[logIntervals, {x, 0, 3}, 
 Ticks -> {({Log10[#], #} & /@ 
    {1, 2, 5, 10, 20, 50, 100, 200, 500, 1000}), None}]