fig, ax = plt.subplots()
ax.loglog(x, y, basex=np.e, basey=np.e)

def ticks(y, pos):
    return '$e^{:.0f}$'.format(np.log(y))
ax.xaxis.set_major_formatter(mtick.FuncFormatter(ticks))
ax.yaxis.set_major_formatter(mtick.FuncFormatter(ticks))
plt.show()
pyplot: loglog() with base e