How can I draw a log-normalized imshow plot with a colorbar representing raw the raw data

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Yes, there is! Use LogNorm. Here is a code excerpt from a utility that I wrote to display confusion matrices on a log scale.

from pylab import figure, cm
from matplotlib.colors import LogNorm

# C = some matrix
f = figure(figsize=(6.2, 5.6))
ax = f.add_axes([0.17, 0.02, 0.72, 0.79])
axcolor = f.add_axes([0.90, 0.02, 0.03, 0.79])

im = ax.matshow(C, cmap=cm.gray_r, norm=LogNorm(vmin=0.01, vmax=1))

t = [0.01, 0.1, 0.2, 0.4, 0.6, 0.8, 1.0]
f.colorbar(im, cax=axcolor, ticks=t, format="$%.2f$")

f.show()

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