import matplotlib.pyplot as plt
import numpy as np
lt = np.linspace(np.log(1000), 5000)
plt.plot(lt, 5 + 2*np.sqrt( lt / 500), 'k-')
np.seterr(all="ignore") # To prevent numpy warnings due to very large numbers.
plt.plot(lt, 10 + 2*np.sqrt(lt / (np.exp(lt) - 500) ), 'r-')
plt.xlabel('log(t)')