Does Subversion have a keyword that behaves like $Log$ in CVS?

No, there is no Subversion equivalent for the $Log$ keyword in CVS. If you want to retrieve a log for a specific file in Subversion, you can run svn log your-file-name or svn log url-to-your-file.

$Log$ is difficult once you start merging changes between branches. You're likely to get conflicts, which cannot be resolved automatically because of the nature of this keyword.

Subversion log messages are mutable - they can be changed by setting the svn:log revision property. So the expansion of $Log:$ in any given file could be out of date. Update may need to retrieve the appropriate log message for each occurrence of the $Log:$ keyword, even if the file that contained it was not otherwise updated.