MySQL is still more popular than MongoDB if you look at Google Trends. I've also added MariaDB, which is pretty much a copy from MySQL and its features, and PostgreSQL, which is also a popular relational database.

This is a very good article for comparing MySQL to MongoDB and which one you should use: MongoDB vs MySQL: A Comparative Study on Databases.

If you just want to learn and you have the time, I would opt for using both MySQL and MongoDB. For example using MySQL for most of the site content and MongoDB for saving log messages. As you get more and more logs you start to see the benefits from MongoDB's faster document fetching.

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