Let's say that I have a problem of linear classification with $K$ classes. The logistic regression model is:
$$\log\frac{P[Class=1|X =x]}{P[Class = K|X=x]} = \beta_{10}+\beta_1^Tx$$ $$...$$ $$\log\frac{P[Class=K-1|X =x]}{P[Class = K|X=x]} = \beta_{(K-1)0}+\beta_{K-1}^Tx$$
so I have :
$$ 1) P[Class=k | X=x] = \frac{e^{\beta_{k0}+\beta_k^Tx}}{1+\sum_{l=1}^{K-1}e^{\beta_{l0}+\beta_l^Tx}}$$ $$2) P[Class=K | X=x] = \frac{1}{1+\sum_{l=1}^{K-1}e^{\beta_{l0}+\beta_l^Tx}}$$
I am not able to understand how to calculate the last two probabilities $1)$ and $2)$. Could you give me any advice on how I should do it?