Not sure about what motivated Cees Elzinga, but the effect of the logarithm is to smooth the impact of the exponential increase of the number of subsequences that can be extracted from the sequence of distinct successive states (DSS).
For an illustration of the dramatic increase of the number of subsequences, see answer in http://stackoverflow.com/questions/20718879/strange-number-of-subsequenceshttps://stackoverflow.com/questions/20718879/strange-number-of-subsequences.
Without the log, the Turbulence would evolve exponentially with the "complexity" of the sequence, which is undesirable.
