Solve a logarithmic equation by NSolve

If you want real solutions, try adding the domain Reals:

NSolve[1 - Abs[x] Log[Abs[(1 + x)/x]] == 0.84, x, Reals]
(*  {{x -> -0.401725}, {x -> -0.0570327}, {x -> 0.0537759}}  *)

Over the Complexes, there are dimension-1 components, which may be why NSolve balks. It probably ought to give an error message, so consider reporting it to WRI.

ContourPlot[
 1 - Abs[x] Log[Abs[(1 + x)/x]] == 0.84 /. x -> a + I b // 
  Evaluate, {a, -1, 1}, {b, -1, 1}]

You should give Mathematica a hint, "what" you are looking for:

NSolve[1 - Abs[x] Log[Abs[(1 + x)/x]] == 0.84, x, Reals]
(*{{x -> -0.401725}, {x -> -0.0570327}, {x -> 0.0537759}}*)