Logistic regression solves a fundamental problem that linear models can't handle: predicting binary outcomes with valid probabilities. While linear regression can output impossible values like -60% or 300%, logistic regression uses a sigmoid function to squish any linear combination of features into a clean probability between 0 and 1.

The core mechanism involves calculating a linear score (z) and passing it through the sigmoid function σ(z) = 1/(1+e⁻ᶻ). This creates a smooth S-curve where z=0 gives exactly 50% probability. For classification, you simply predict class 1 when P(y=1) ≥ 0.5, or adjust this threshold based on your precision-recall tradeoff needs.

Unlike linear regression's squared error, logistic regression maximizes likelihood using cross-entropy loss, which creates a convex optimization surface. This means gradient descent reliably finds the global optimum. The resulting coefficients are interpreted as log-odds changes: each unit increase in a feature multiplies the odds by e^β, making the model highly interpretable for business decisions.