Try something like this:
from matplotlib import pyplot as plt
import numpy as np
Xs = np.linspace(-2.0, 2.0)
Ys = 10.0**Xs
plt.plot(Xs, Ys)
plt.yscale("functionlog", functions=(
lambda x: np.log10(x + 1.0),
lambda x: 10.0**x - 1.0))
plt.show()
