How flexible is the link between objective function and output layer activation function?
It is not so much which activation function which you use which determines which loss funtion you should use, but rather what the interpretation you have of the output is.

If the output is supposed to be a probability, then log-loss is the way to go. 

If the output is a generic value then mean squared error is default way to go. So for example, if your output was a grey scale pixel with grey-scale labelled by a number from 0 to 1, it might make sense to use a sigmoid activation function with a mean squared error objective function.