![]() The Mann-Whitney U-statistic counts the number of times a value in one sample exceeds a value in the other. a scale-shift is a location-shift in the logs), it has power against any alternative that makes $P(X>Y)$ differ from $\frac$. Not only can the Mann-Whitney deal with transformed-location shifts very well (e.g. In particular, it is able to test whether the probability of values in the first group are larger than the values in the second group, which is quite a general alternative that sounds like it's related to your original question. The Mann-Whitney doesn't require equal variances unless you're specifically looking for location-shift alternatives.
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