import matplotlib.pyplot as plt
import numpy as np
import scipy.optimize as sco

x = np.array([180, 300, 560, 1000, 2000, 4999, 10000, 20000])
y = np.array([826.68, 1169.74, 1709.03, 2255.51, 2827.74, 3302.12, 3486.19, 3681.87])

def log(bla):
    a, b, c = bla
    return a * np.log(c*x) + b - y

def sigmoid(bla):
    d, a, c, b = bla
    return d + (a-d)/(1+(x/c)**b) - y

res_log = sco.least_squares(log, [1., 0., 10.], method='lm').x
res_sigmoid = sco.least_squares(sigmoid, [4000., 0., 1., 1.], method='lm').x

plotty = np.geomspace(100, 30000)
plt.plot(x, y, 'rx')
plt.plot(plotty, res_log[0]*np.log(res_log[2]*plotty) + res_log[1], 'g-', label='log')
plt.plot(plotty, res_sigmoid[0] + (res_sigmoid[1] - res_sigmoid[0]) / (1+(plotty/res_sigmoid[2])**res_sigmoid[3]), 'b--', label='sigmoid')
plt.legend()
plt.show()