When can you reorder log operations?

Question

For example, you can reorder a softmax + nl (negative likelihood) to log_softmax + nll (negative log-likelihood)

Essentially changing log(softmax(x)) to softmax(log(x))

However, what are the rules to reordering logging of things?

Answer 1

In general, you cannot reorder like that. The example you give is a very special case that only works because softmax is based on the exponential function, which is the inverse function of the natural logarithm.