Why only logging? Why not Database connection? I have been using one database for my simple small CMS and I guess anybody, any fool like me, will go for a singleton. You are not going to again and again open a db connection for 2 queries at line number m and m+1. Its better to have a single connection and pass it over. That’s it. Its the benefit part. Now the coupling. Yes my user class is totally dependent on the database class. Anything wrong or any small change in mydb class means check user class also.
The only point I can make is as de-normalization helps and so does singleton. ;-)