A calculus problem by zhida wang

I used Newton Raphson iteration. Start by taking the natural logarithm of both sides of the first equation.

$x \, \ell n x = \ell n 2$

Define a function $f$ and its derivative:

$f = x \, \ell n x - \ell n 2 \\ f' = \frac{df}{dx} = 1 + \ell n (x)$

Starting with an initial guess, update $x$ as follows:

$x_k = x_{k - 1} - \frac{f_{k-1}}{f'_{k-1}}$

Code and results below:

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import math
import random

x = 1.0 + random.random()

for j in range(0,100):

    f = x*math.log(x) - math.log(2.0)

    fp = 1.0 + math.log(x)

    x = x - f/fp

print x
print (x**x)

#>>> 
#1.55961046946
#2.0
#>>>