The second form above illustrates that the
logarithm of the geometric mean is the weighted arithmetic mean of the logarithms of the individual values.
If all the weights are equal, the weighted geometric mean simplifies to the ordinary unweighted geometric mean.[1]
References
^
abSiegel, Irving H. (June 1942), "Index-number differences: geometric means", Journal of the American Statistical Association, 37 (218): 271–274,
doi:
10.1080/01621459.1942.10500636