I don't believe I stated that exponentiation produces the mean of the non-logged data. Assuming that the log-transformation is warranted due to extreme skew or outliers then the mean of the non-logged data will not give useful information (i.e., is not a typical value of the distribution). Log transformation "normalizes" the distribution - but the log-mean is not a meaningful value (in terms of the original unit of measurement of the data), hence makes sense to take the anti-log to bring the mean value back to the original unit of measurement. The new value (which as you correctly point out is the geometric mean) is a "normalized mean" and is, of course, different from the original mean... but assuming that the outliers in the original distribution are results of anomaly, it is a better representation of the central location. Why is that difficult?
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