We present a local convergence analysis of an iterative method of convergence order six and efficiency index 1.8171 in order to approximate a locally unique solution of a nonlinear equation. In earlier studies such as [Nazeer. W, Modified Halley's method for solving nonlinear equations with convergence order six and efficiency index 1.8171, Appl. Math. Comput., 283,(2015), 57-69.] the convergence order of these methods was given under hypotheses reaching up to the fourth derivative of the function although only the first derivative appears in these methods. In this paper, we expand the applicability of these methods by showing convergence using only the first and second derivatives. Moreover, we compare the convergence radii and provide computable error estimates for these methods using Lipschitz constants.