I think thats the whole point why we use log softmax instead of softmax. i.e., numerical stability.
If we recall the softmax formula, It involves exponential powers. When we have large numbers (as in the array you mentioned), due to limited numerical precision of our machine, the softmax just kills the precision of numbers.
When there is exponentiation involved, log comes to the rescue to avoid the numbers blowing up. Also, In our case, softmax is mostly used in conjunction with CrossEntropyLoss function which needs log likelihood (log probability). By keeping these two reasons in mind, the researchers had come up with a clever trick to directly take log to avoid numerical precision errors.
A glimpse of softmax-cross-entropy derivation:

  
      DeepNotes – 28 May 17
  
  
    

Classification and Loss Evaluation - Softmax and Cross Entropy Loss 35

Lets dig a little deep into how we convert the output of our CNN into probability - Softmax; and the loss measure to guide our optimization - Cross Entropy.