See Bernoulli's probability distribution

P (Y=y)= py (1-p)1-y = (p/1-p)y (1-p) = eyln(p/1-p)+(1-p) y=0, 1

So the Bernoulli distribution can easily be converted to an exponential distribution.

We call ln(p/1-p) the natural parameter of exponential distribution.

Now in a logistic regression model, this natural parameter is placed in the Dependent Variable position, ie

ln(p/1-p) = y = b0 +b1X1 + b2X2 + .... + bkXk y=0, 1

Pay attention to y=0, 1

Dependent variable is not a categorical dependent variable, it can only accept two values of zero and one.

Logistic regression is originally a regression technique. By including a decision rule, it is transformed into a classifier. It is more successful in classification problems but it is also used in regression problems.

I agree with Serkan Balli's point

Logistic regression is the appropriate regression analysis to conduct when the dependent variable is dichotomous (binary).  Like all regression analyses, the logistic regression is a predictive analysis.  Logistic regression is used to describe data and to explain the relationship between one dependent binary variable and one or more nominal, ordinal, interval or ratio-level independent variables.

On the other hand, the logistic classification model (or logit model) is a binary classification model in which the conditional probability of one of the two possible realizations of the output variable is assumed to be equal to a linear combination of the input variables, transformed by the logistic function.

The way I see it, logistic regression is a classifier that uses regression techniques. The output of the logit function used in logistic regression is indeed continuous (probability of log odds). However, you can impose a threshold to these probabilities and thus predict categories. Thus, you successfully used a regression technique to predict classes.