Interpretation of the back-transformed value as a median requires that the residuals of the log-transformed response have a symmetric distribution. The residuals do not have to be normally distributed, just symmetric.
It should be noted that any transformation which achieves a symmetric distribution for the residuals will result in a back-transformed estimate which is interpretable as the median, assuming that the back-transformation is applied to an estimate of the mean. The mean of the log-transformed (or square-root transformed, or inverse, or ...) data is an estimate of the 50th percentile of the transformed response as long as the transformed data are symmetric. Back-transforming the 50th percentile of the transformed response will result in an estimate of the 50th percentile on the original scale.
... View more