Logistic Regression : Solving the cross-entropy cost function analytically
If you use just one neuron (linear->sigmoid), you can find the minimum with too much small error or maybe reach to the minimum using approaches like gradient descent or other optimization algorithms. The reason for finding the minimum is that cross entropy is a non-convex shape, but if you use sigmoid function as the activation function of logistic regression, the cross entropy cost function becomes convex and it is easy to find the only global minimum.