import matplotlib.pyplot as plt
import numpy as np

x = np.linspace(1, 40, 100);
y = np.linspace(1, 4, 100);

# Actually plot the exponential values
plt.plot(x, np.e**y)
ax = plt.gca()

# Set x logaritmic
ax.set_xscale('log')

# Rewrite the y labels
y_labels = np.linspace(min(y), max(y), 4)
ax.set_yticks(np.e**y_labels)
ax.set_yticklabels(y_labels)

plt.show()

plt.figure()
ax = plt.subplot(111)
ax.set_yscale('log')

class ExponentialScale(mscale.ScaleBase):
    name = 'expo'
    base = 1.1
    logbase = math.log(base)

def __init__(self, axis, **kwargs):
    mscale.ScaleBase.__init__(self)
    self.thresh = None #thresh

def get_transform(self):
    return self.ExponentialTransform(self.thresh)

def set_default_locators_and_formatters(self, axis):
    # I could not get LogLocator to do what I wanted. I don't understand
    # the docs about "subs" and the source was not clear to me.
    # So I just spell out the lines I want:
    major = [1, 5, 10, 12, 14, 16, 18, 20, 25, 28, 30] + range(31,60)
    axis.set_major_locator(ticker.FixedLocator(major))

class ExponentialTransform(mtransforms.Transform):
    input_dims = 1
    output_dims = 1
    is_separable = True

    def __init__(self, thresh):
        mtransforms.Transform.__init__(self)
        self.thresh = thresh

    def transform_non_affine(self, a):
        res = ma.power(ExponentialScale.base, a)
        return res

    def inverted(self):
        return ExponentialScale.InvertedExponentialTransform(self.thresh)

class InvertedExponentialTransform(mtransforms.Transform):
    input_dims = 1
    output_dims = 1
    is_separable = True

    def __init__(self, thresh):
        mtransforms.Transform.__init__(self)
        self.thresh = thresh

    def transform_non_affine(self, a):
        denom = np.repeat(ExponentialScale.logbase, len(a))
        return np.log(a) / denom

    def inverted(self):
        return ExponentialScale.ExponentialTransform(self.thresh)

...
ax = plt.subplot(111)
ax.plot(Xs,Ys,color='blue',linewidth=2)
....
xlabs = [pow(10,i) for i in range(0,6)]
ax.set_xticklabels(xlabs)
ax.set_xticks(xlabs)