Add 1 to each x value, then take the log:
import matplotlib.pyplot as plt
import numpy as np
import matplotlib.ticker as ticker
fig, ax = plt.subplots()
x = [0, 10, 100, 1000]
y = [100, 20, 10, 50]
x = np.asarray(x) + 1
y = np.asarray(y)
ax.plot(x, y)
ax.set_xscale('log')
ax.set_xlim(x.min(), x.max())
ax.xaxis.set_major_formatter(ticker.FuncFormatter(lambda x, pos: '{0:g}'.format(x-1)))
ax.xaxis.set_major_locator(ticker.FixedLocator(x))
plt.show()
Use
ax.xaxis.set_major_formatter(ticker.FuncFormatter(lambda x, pos: '{0:g}'.format(x-1)))
ax.xaxis.set_major_locator(ticker.FixedLocator(x))
to relabel the tick marks according to the non-log values of x.
(My original suggestion was to use plt.xticks(x, x-1), but this would affect all axes. To isolate the changes to one particular axes, I changed all commands calls to ax, rather than calls to plt.)
matplotlib removes points which contain a NaN, inf or -inf value. Since log(0) is -inf, the point corresponding to x=0 would be removed from a log plot.
If you increase all the x-values by 1, since log(1) = 0, the point corresponding to x=0 will not be plotted at x=log(1)=0 on the log plot.
The remaining x-values will also be shifted by one, but it will not matter to the eye since log(x+1) is very close to log(x) for large values of x.
