Help in understanding the maths behind Logistic Regression

Question

I am following the lecture notes available https://www.stat.cmu.edu/~cshalizi/uADA/12/lectures/ch12.pdf

I cannot understand how Eqs 12.4 and 12.5 come,

• why the Bernoulli probability has $1-p(x)$ in the denominator,
• how come $p(x) = \exp(\beta + \beta^Tx)$
• and how $log \frac{p(x)}{1-p(x)}$ evaluates to $\beta + \beta^Tx$.

In general $\beta$ is the parameter of the model but I don't quite follow how come the log expression evaluates to it. Is there some mathematical formula which is skipped that is used to evaluate the log expression? This is crucial for me to know as these values are substituted in eq 12.10 where $p(x) = \exp(\beta + \beta^Tx)$