Develop a set of detailed lecture notes covering the following topics related to logistic regression:

$1)$Introduction to logistic regression for binary classification tasks.

$2)$Implementation of the logistic regression algorithm using a real$-$world dataset $($e$.$g$.$ for predicting customer churn or spam detection$).$

Explain the key concepts, mathematical formulas, and steps involved in each of these topics. Be sure to include example code or pseudocode demonstrating the implementation. The notes should be comprehensive enough to serve as a study guide for students learning about logistic regression.