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Does logistic regression only solve binary classification problems? How can logistic regression solve multiple-class problems?

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No, multiclass classification is also possible. Try reading up on 'One vs All' multiclass classification.

https://en.wikipedia.org/wiki/Multiclass_classification

Multiclass, One vs All classification in the case of logistic regression:

https://www.coursera.org/learn/machine-learning/lecture/68Pol/multiclass-classification-one-vs-all

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