Problem 4(Newton's method) Consider the nonlinear equation \[ f(x)=\ln (x)-1\] a) Find the root analytically. b) Show that Newton's method is a special case of fixed-point iteration. c) Explain briefly why Newton's method will converge to the root, assuming the initial guess is sufficiently close. Justify your answer mathematically. d) With \( x_{0}=3\), perform three Newton iterations by hand. e) At each iteration, compute the relative error between successive iterates.