Never mind, I found the answer myself:
This is not possible with standard Unix-Permissions.
However, that does not mean it's not possible on Linux.
Access Control Lists (ACLs) are applied to files and directories. ACL
behavior is defined by IEEE's POSIX 1003.1e draft and supports
control/access of signals, TCP/IP ports (below 1024), raw sockets, ...
ACLs are an addition to the standard Unix file permissions (r,w,x,-)
for User, Group, and Other for read, write, execute and deny
permissions. ACLs give users and administrators flexibility and direct
fine-grained control over who can read, write, and execute files.
The Linux 2.6 kernel (beginning with Fedora Core 2) supports ACLs for
EXT2, EXT3, XFS, JFS, and ReiserFS file systems.
And from http://www.linux-tutorial.info/modules.php?name=ManPage&sec=1&manpage=setfacl
setfacl -m u:bind:rw /var/log/DESIRED_FILENAME_HERE
Also, modern Linux is using ext4.
So note this:
You need to remount the mount point -- in this specific case, the root
/ filesystem, not just the desired directory. In other words, try
sudo mount -o remount,acl /
For future, change the line in /etc/fstab to
UUID=66eeee3e-b860-41b0-abf7-074c0e08420e / ext4 relatime,acl,errors=remount-ro 0 1