I think both of log-loss and f-score can handle the unbalanced data and the f-score is normalized and more interpretable than log-loss. However, is there any advantage for log-loss over f-score?
The log-loss is a type of loss function. Your model will optimize it (e.g. minimizing) for reaching your goal. Usually it needs to be differentiable, since you could need to calculate the derivative for getting the max or min (these details depend from your optimization algorithm).
The f-score is a metric. A metric is used to evaluate the performance of your model, in this case considering the harmonic mean between precision and recall, but it's not involved in the optimization process and they are not required to be differentiable (so you can't use them for training your model).