Open certain local files in the browser, not in external application

I have a html page with several links to some files with different names.
The files are all simple text files but they do not have the ".txt" suffix.

When I open the page with firefox and try to open the linked files, it
behaves differently depending on the suffix of the file. Some files are
opened directly within the browser window (e.g. a file named "X.output")
but for other files (e.g. "Y.log" or "Z.diff"), firefox wants to save the
file to disk or have an external application that opens it.

I can change the external application, but I can not tell firefox to open
the files directly.
Note, that all the files are local files and no web-server is involved.

How can I tell firefox to open these files directly in the browser window?
Can I change the behaviour for all "*.log" files, for example?