Comparing survival functions with the log-rank test in the presence of dependent censoring can produce an invalid test result. We extend our previous work on the estimation of the survival function using prognostic variables to adjust for dependent censoring to the comparison of two survival distributions. In these estimators, the weights of a censored individual is redistributed among either a subset of patients in the risk set or all patients in the risk set but giving more weight to patients having smallest distances from the censored subject. The distance is based on two risk scores obtained from two working models, one for the failure time and one for the censoring time. Based on the estimators, we suggest a weighted log-rank test to compare two survival distributions. A simulation study compared performance of our method with the analysis of the observed data without using auxiliary variables and with a recent method based on multiple imputation (KMIB method). With appropriate parameters, the weighted log-rank approach provides sizes of the test comparable to the nominal value but higher powers than the two other methods. The method is illustrated with data from a breast cancer study.