Log and register artifacts are a subgroup of project artifacts that has continuously updated project management documents as a result of progressive elaboration.
These are generally created at a specific point of a project and then used for many tailored processes. This usage brings about the unavoidable need for updating the document continuously. The terms "log" and "register" has very close meanings. So there is nothing wrong if you confuse their names while mentioning them, or do not be surprised if you see them by different names. However, in project management exams it is recommended to learn the common usage of the term in the source you use for that specific exam.
The commonly used logs and registers for all development approaches are: