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/wi

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.