Multi-class classification is a generalization of logistic regression wherein we are dealing with binary classification. The latter problem is a setting where a number should be mapped to either $0$ or $1$. Hence, Logistic regression needs to convert the output of a neural network ($\hat{y}$) to to either $0$ or $1$ to decide. Therefore it use the sigmoid function defined as $$\sigma(\hat{y})=\frac{e^\hat{y}}{1+ e^\hat{y}}\tag{1}$$ where $\hat{y} \in \mathbb{R}$ is the output of the network.
On the other hand, Multi-class classification uses the softmax function to decide which is defined as
$$\text{softmax}(\hat{\textbf{y}})=\frac{e^{\hat{\textbf{y}}_i}}{\sum_{i=1}^{n}e^{\hat{\textbf{y}}_i}}\tag{2}$$ where $\hat{\textbf{y}} \in \mathbb{R}^n$ is the output of the network.
Question: How can we can play with $(1)$ to get $(2)$ algebraically or vice versa? If we start with $(1)$ how one can get rid of $1$ in denominator? or if we start with $(2)$ how we can generate $1$ in the denominator where $n=2$.