Move log file without taking database offline
There is no way to do this with an online database.

When you move a database file (ALTER DATABASE ... MODIFY FILE), you even get the following message:


  The file "YourFile" has been modified in the system catalog. The new path will be used the next time the database is started.





  The normal way of doing this would be to detach the DB, move the log file then reattach the db.


That's not the "normal" or accepted way I would do it.  To move database files, I do the following:


Run an ALTER DATABASE command to change the location of the file(s)
Take the database offline
Physically move the file(s) to the new location specified in step #1
Bring database online


See this reference on TechNet: Move User Databases