Although I will confess to nitpicking as I've never seen anyone using decimal OoM.

The unit of "deciBel" (dB) is used for measuring sound pressure and electrical signal levels.

It is a logarithmic scale, so 24dB = 10

^{2.4} See:

http://en.wikipedia.org/wiki/DecibelIn the equal-tempered musical scale, the frequencies also follow a logarithmic scale (and most other musical scales are approximately logarithmic, at least at the level of an octave). This is a non-decimal exponent, so if you were to express it as powers of 10, it would be a fractional exponent.

http://en.wikipedia.org/wiki/Scale_(music)

Other examples are the brightness of stars, which is a logarithmic scale, with a non-decimal base. It is common for astronomers to quote magnitudes with 1 decimal place.

http://en.wikipedia.org/wiki/Apparent_magnitudeIn geology, the Richter scale for earthquakes is a base 10 logarithmic scale, and fractional magnitudes are often quoted.

http://en.wikipedia.org/wiki/Richter_magnitude_scaleThe Volcanic Explosivity Index for supervolcanoes is logarithmic.

http://en.wikipedia.org/wiki/Volcanic_explosivity_indexMoh's scale of mineral hardness is approximately logarithmic (except diamond was way off the scale, and was arbitrarily assigned 10 on the scale).

http://en.wikipedia.org/wiki/Mohs_scale_of_mineral_hardnessIn chemistry, the pH scale is logarithmic, and it is easy to measure pH to one decimal place.

http://en.wikipedia.org/wiki/PH#Definition_and_measurementMany human senses have a logarithmic response. A quantity that spans many orders of magnitude is often measured most easily with a logarithmic scale; if precision is required, a fractional order of magnitude is often used.

And back near our initial topic: X-Class Solar Flares are also categorized according to a somewhat bizarre logarithmic scale. (Lower class flares have one decimal place, but this is effectively scientific notation, rather than a decimal order of magnitude.)

http://en.wikipedia.org/wiki/Solar_flare#Classification