Newton's method with initial approximation

Question

Use Newton's method with initial approximation x₁=1 to find the third approximation of the root (i.e. solution) of the equation ln(x)=1/(x-3) nearest to the initial approximation. Round your answer to 6 decimal places.

Answer 1

x₁ = 1; f(x) = lnx - 1/(x - 3); f'(x) = 1/x + 1(x - 3)² = (x² - 5x + 9)/[x(x - 3)²]

x₂ = x₁ - f(x₁)/f'(x₁) = 1 - (1/2)/(5/4) = 0.6

x₃ = x₂ - f(x₂) - f(x₂)/f'(x₂) = 0.6 - (-.0511656) = 0.651166