A note on a functional equation (Q5903221)

Recently \textit{B. Ross} [Indian J. Pure Appl. Math. 14, 395-397 (1983; Zbl 0556.26004)] solved the equation \[ (1)\quad -K= \ln x+\sum^{\infty}_{n=1}(-1)^{n-1}\frac{x^ n}{n!},\quad K\text{ (constant) } >0, \] explicitly. Since the solution is given in the form of an infinite series (whose convergence ''depends upon the magnitude of K''), the usefulness of explicit solution is as limited as that of an existence proof a constructive nature, for example via Banach's contraction principle. The purpose of this note is to prove the existence of a unique solution of (1), which can be approximated by an iterative procedure.