## Softmax Activation Function

The softmax activation function is a non-linear function that is commonly used in the output layer of neural networks for multi-class classification problems. It takes a vector of real numbers as input and outputs a vector of probabilities, where the probabilities sum to 1. This means that the softmax function can be used to represent a probability distribution over the possible output classes.

The softmax function is defined as follows:

```
softmax(x) = [exp(x_1) / sum(exp(x_i)) for x_i in x]
```

where x is the vector of input values and exp(x) is the exponential function. The softmax function first exponentiates each element of the input vector, which makes them all positive. It then divides each exponentiated value by the sum of all the exponentiated values. This ensures that the output vector of probabilities sums to 1.

The softmax function can be interpreted as a way of normalizing the output of a neural network so that it represents a probability distribution. This is useful for multi-class classification problems, where the goal is to predict the probability of a given input belonging to each of the possible output classes.

## Example of how the softmax function can be used to classify images

Suppose we have a neural network that has been trained to classify images of cats and dogs. The output layer of the neural network will have two neurons, one for each output class. The softmax function will be applied to the output of these neurons to produce a vector of two probabilities, one for the probability of the input image being a cat and one for the probability of the input image being a dog. The neuron with the highest probability will be the predicted class for the input image.

The softmax function is a powerful and versatile activation function that is commonly used in neural networks for multi-class classification problems. It is easy to implement and understand, and it produces outputs that are easy to interpret.