If I understand your intent the short answer I believe is NO.
You can specify using Proc Printo a file to write the LOG to for a session but you would have to specify it in each of your sessions and a separate file name to write too for each.
Harder to manage would be the SAS system opton ALTLOG as that would need to be set when each session is invoked (command line option for example not just clicking on your desktop icon for a second session).
And log output may not have what you want for every script. Data steps with inline datalines or cards statement data do not show the lines of data in the log.