I'm studying Logistic Regression, there is a thing that I'm not able to understand.

I know what the Logistic Function is, I studied it. But I don't understand why the logistic function is equal to the dot product between $w$ and $x$.

$$\log \left(\frac{p}{1-p}\right)=\bar{\omega }\cdot \bar{x}$$

For me these are two completely different concepts, I don't understand why are related in this manner. Can you help me?

It is just a matter of modelling.

Note that $w \cdot x = \sum_{i=1}^n w_i x_i$.

For example in linear regression (ignore intercept term for now, we can always insert the intercept term by including $1$ as a feature), we try to fit a model of the type of $y(x) = \sum_{i=1}^n w_ix_i$, however, the problem with this when the dependent variable is that $y$ can take values beyond $0$ and $1$.

Hence, we consider another model $F(p) = \sum_{i=1}^n w_ix_i$ where $F(p)$ can take any real value and $F$ is a function which takes value from $(0,1)$ and be able to map it to $\mathbb{R}$. $F(p)$ can be chosen to be $\log \left( \frac{p}{1-p} \right)$, which is an increasing function which map $(0,1)$ to $\mathbb{R}$.

Logistic regression is a statistical method for analyzing a dataset in which there are one or more independent variables that determine an outcome. The outcome is measured with a dichotomous variable in which there are only two possible outcomes. You can think like entropy in decision trees. If probability of situation is binary there is no relationship with outcome.

The function that is included to your post about theory. In an application you are going to use this term for it.

$w \cdot x = \sum_{i=1}^n w_i x_i$ or $F(p) = \sum_{i=1}^n w_ix_i$

And it will give you a curve shape like half of Gaussian curve