Change database used for SQL Server Integration Services logging
Blowing Kin's comment up to an answer...

Logging in the package deployment model (2005, 2008/2008R2 and potentially 2012/2014/2016 depending on how the project is defined) is done through an explicit request within each package. 

A package looks something like this. At the bottom, you see 3 entries under Connection Managers: msdb, SO_34033143 and NotMsdb. The middle is a flat file connection manager, the other two are OLE DB connection managers. All a Connection Manager does is define the specifics of the entity the package is going to work with: this database on that server logging in as user123. A CSV named XYZ.csv, etc. 



It's important to note that the name given to a Connection Manager may have no bearing on what it is actually connecting to. For example, the one named "msdb" is actually pointing to tempdb because I'm lazy.

If you right click in the grey background between the Data Flow Task and the Connection Managers, you should get a context sensitive menu and pick Logging



In the resulting screen, you should see a menu pop up that has on the right hand side "SSIS Log Provider for SQL Server" checked. The third column indicates the connection manager that is being used to log SSIS.



What you'll want to do and again, this is for every package, go in and change the connection manager from the thing that is pointing to msdb and either use an existing connection manager or add a dedicated connection manager into the packages.



Personally, in my pre-2012 environments, I advocated for an SSISConfig/SSIS_Config database. I used that for logging and configuration across a given tier/class of servers (dev/stage/prod).

If you're handy with scripting, you can programatically open every SSIS package and change the logging provider but it's not for the faint of heart. I don't believe I have that code online but if you're super desperate, ping me via comment and I'll see if I can find that hard drive.