Logarithm of the factorial

Use Stirling's Approximation to minimize computation and bound the search between $10^{10}$ and $10^{11}$ . Also, use the fact that $log_x {n} = \frac{log(n)}{log(x)}$ .

Solution based on the Stirling's approximation in python 3.4:

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from math import log

def f(x):
    return (1 - x)/log(x) + x + 1/2
def solve(c):
    "Logarithm of the factorial"
    n = 2
    while f(n) < c:
        n *= 2
    m = n//4
    while m > 1:
        if f(n - m) > 11814375113: n = n - m
        m = m//2
    return n

print(solve(11814375113))

i will upload the answer soon as possible. i will be uploading a snap.