What is the advantage of using log softmax instead of softmax
There are a number of advantages of using log softmax over softmax including practical reasons like improved numerical performance and gradient optimization. These advantages can be extremely important for implementation especially when training a model can be computationally challenging and expensive. At the heart of using log-softmax over softmax is the use of log probabilities over probabilities, which has nice information theoretic interpretations.

When used for classifiers the log-softmax has the effect of heavily penalizing the model when it fails to predict a correct class. Whether or not that penalization works well for solving your problem is open to your testing, so both log-softmax and softmax are worth using.