It's not quite what you are after, but a great example of logarithms is in the news right now - earthquakes are measured on a log scale.

Although the official measurement of quakes is done on the Moment magnitude Scale, ( http://en.wikipedia.org/wiki/Moment_magnitude_scale ), the media still like to report using the old Richter magnitude scale ( http://en.wikipedia.org/wiki/Richter_magnitude_scale ) - probably because more people have heard of it.

The energy released by an earthquake, which closely correlates to its destructive power, scales with the power of 1.5 of the shaking amplitude. A difference in magnitude of 1.0 is equivalent to a factor of 31.6 in the energy released; a difference in magnitude of 2.0 is equivalent to a factor of 1000 in the energy released.

So the difference between the 8.9 size quake in Japan and the recent 6.3 size quake in New Zealand is 8.9-6.3=2.6, so the difference in destructive power is (inverse log 2.6) to the power of 1.5 - which gives an answer of nearly 8000x the destructive force in the recent Japan quake!

By the way, inverse log of x = 10 to the power of x.