The key here is the cax kwarg to colorbar. You'll need to create an inset axes, and then use that axes for the colorbar.
As an example:
import numpy as np
import matplotlib.pyplot as plt
npoints = 1000
x, y = np.random.normal(10, 2, (2, npoints))
fig, ax = plt.subplots()
artist = ax.hexbin(x, y, gridsize=20, cmap='gray_r', edgecolor='white')
# Create the inset axes and use it for the colorbar.
cax = fig.add_axes([0.8, 0.15, 0.05, 0.3])
cbar = fig.colorbar(artist, cax=cax)
plt.show()

If you wanted to get fancy and more precisely match things (Note: I'm using hexbin here, which doesn't support log axes, so I'm leaving that part out.)
import numpy as np
import matplotlib.pyplot as plt
npoints = 1000
x, y = np.random.normal(10, 2, (2, npoints))
fig, ax = plt.subplots()
artist = ax.hexbin(x, y, gridsize=20, cmap='gray_r', edgecolor='white')
cax = fig.add_axes([0.8, 0.15, 0.05, 0.3])
cbar = fig.colorbar(artist, cax=cax)
ax.spines['right'].set(visible=False)
ax.spines['top'].set(visible=False)
ax.tick_params(top=False, right=False)
cbar.set_ticks([5, 10, 15])
cbar.ax.set_title('Bin Counts', ha='left', x=0)
cbar.ax.tick_params(axis='y', color='white', left=True, right=True,
length=5, width=1.5)
cbar.outline.remove()
plt.show()
