from math import log
def f(x):
    return x*x*(x+2)+log(x) -4

epsilon = 1.0e-8
x=[1,1.5]
while abs(x[-1]-x[-2])> epsilon:
    x1,x2 = x[-2], x[-1]
    x.append(x2-f(x2)*(x2-x1)/(f(x2)-f(x1)))

print(x[-1])

1.117033697609942