The most used loss function in tensorflow for a binary classification?
I think there is some confusion here. Softmax is usually an activation function which you will use in your output layer, and cross-entropy is the loss function that you will use.

Softmax

This activation function outputs the probability for each class, it is defined as

$\sigma(\bf{y_i}) = \frac{e^{\bf{y_i}}}{\sum_{j}{e^{\bf{y_j}}}}$.

For example, in a problem with 2 class labels. Then if we have some outputs from our neural network like $y$=[3, 4] then we can get the probability of each output classes by using the softmax function on these outputs.

import numpy as np

x = [3, 4]
np.exp(x)/np.sum(np.exp(x))



  [0.26894142, 0.73105858]


You will see that these are probabilities and do sum to 1. 

Then we can get the class of the input by seeing which probability is higher. In this case class 2 has a probability of 73.11%, so the predicted class label, $\hat{y} = 1$.

Cross-entropy

Cross-entropy is a loss function that is used for classification tasks. For binary classification it is defined as 

$H(p, q) = -y\log(p) - (1-y)\log(1-p)$.

Let's assume that the real class of the above example is 0, $y=0$. Then we made a mistake and you can see that 

$H(p, q) = -0\log(0.26894142) - (1-0)\log(1-0.26894142) = 0.313$.

That is the loss that is used for backpropagation.