From the geometric interpretation of the Newton method in the scalar case, we see that the method runs faster the lower the convexity of the function involved in the equation to be solved. To measure the convexity of the function, we can use the degree of logarithmic convexity defined in Chap. 1 , which, as we have seen, is a point measure of convexity. We begin the chapter by studying the convergence of the Newton method on the real line and continue looking at the influence that the convexity of the function involved has on the speed of convergence of the method. Next, we do two analyses of the convergence of the method in Banach spaces. For this, we first extend the degree of logarithmic convexity to Banach spaces, and although the geometric sense that it has in the scalar case is lost, we analyze its existence based on the classical Kantorovich conditions. In the first study, the convergence of the method is analyzed based on the type of conditions required to the degree of logarithmic convexity, emphasizing the majorant principle introduced by Kantorovich to study the semilocal convergence of the Newton method, the fixed point techniques for the study of the global convergence, which guarantee the extension of the domain of starting points, together with the use of auxiliary points, and the ideas contributed by Dennis and Schnabel to obtain the local convergence of the method. In the second study, we use the degree of logarithmic convexity under Kantorovich-type conditions. In semilocal convergence studies, we use the theoretical significance of the method to draw conclusions about the existence and uniqueness of solution in some cases, as well as to give a priori error estimates based on the Ostrowski technique.