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How is the log loss function derived?

Loss functions are used to guide the development of machine learning (ML) models and each model type has one or more preferred loss functions. I started learning about them in a more structured manner recently to strengthen my foundation in ML. I started with linear regression and their loss functions and moved on to logistic regression. Logistic regression uses the log loss function which looks much more complicated than the loss functions used in linear regression, and I couldn’t help but wonder why. How is the log loss function derived? Why is it the way it is?