approximating logarithm using harmonic mean

here is function to approximate log10(x+1), (x+1) < ~1.2

a = 1.097
b = 0.085
c = 2.31

ans = 1 / (a - b*x + c/x)

so it should look like that
it works by adjusting harmonic mean to match log10
but the problem is in values of abc

the question is how to get just right a, b and c
and how to make better approximation

i made this code that can give you pretty good approx for a, b, c
but my code wasnt able to make it any better

here is my code:

import numpy as np

a = 1
b = 0.01
c = 2

def mlg(t):
    x = t
    if t == 0:
        x = 0.00000001
    x2 = x*x
    o = a - (b * x) + (c / x)
    return 1/o

def mlg0(t):
    x = t
    if t == 0:
        x = 0.00000001
    x2 = x*x
    o = a - (b * x) + (c / x)
    return o

for i in range(9000):
    n1 = np.random.uniform(0,1.19,1000)
    for i in range(1000):
        n = n1[i]
        o = np.log10(n+1)
        u = mlg(n) - o
        e = u ** 2

        de_da = 0 - 2 * (u) / (mlg0(n) ** 2)
        de_db = de_da * n 
        de_dc = de_da / n

        a -= de_da * 0.00001
        b -= de_db * 0.00001
        c -= de_dc * 0.00001

print(a,b,c)

so do u guys have any ideas of how to generate better values?

i`ve used method alike back propagation in NN but it could not give me values any better

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