Sorry but the formulas you gave for the log loss and cross entropy loss are wrong. The log loss is the same thing as the binary cross entropy loss. The formula you gave for the cross entropy loss is to be used for binary classification and should be called binary cross entropy. The formula you gave for the log loss is plain wrong. When you want to compute the cross entropy for a multi classes classification problem, you need to add up one term (y log(y_hat)) by class

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