Subscript[E, 4] = x/(Exp[x] - 1) - Log[1 - Exp[-x]]; intp = Reduce[Subscript[E, 4] == 1/x, x, Reals]
Plot[{Subscript[E, 4], 1/x}, {x, 0, 3}, Epilog -> {Red, PointSize[0.025], Point[{x, Subscript[E, 4]}] /. {ToRules[intp]}}]