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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. The log loss function is used for training logistic regression models 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?

Why is the harmonic mean used for the F-score?

Precision and recall are common metrics used for assessing the performance of binary classification models. Due to their definition, however, there is usually a trade-off point between precision and recall. At this trade-off point the higher the precision, the worse the recall, or vice versa. The F-score (also known as the ‘F1-score’) is an alternative metric that represents both the precision and recall. It is often defined as the harmonic mean between precision and recall. What is the harmonic mean and why is it used in the F-score?