Store logging in separate or existing database?
I would go for the same database, for the reasons you state in your question. If you're worried about performance or growth, create a new "logging" filegroup and put the logging table(s) on that. If things grow out of hand in the future, you can move the .ndf (the physical file that contains the new filegroup) to a different physical drive.

Even if you stick to the same server and add a new database, the effect is similar. You don't have to create a new connection in your client application - all you have to do is reference the table by a three-part name, [database].[schema].[table].

Edit: also, consider if the logged information should "follow" the rest of the database, or if it's specific to a certain server or environment.