If there's no DLS subfolder in your ~/Documents folder and you cannot find DLS.log anywhere it probably means that DLS doesn't have permissions to initialize its "file system".
The first thing DLS does when starting up is to figure out what's the installation path (readlink("/proc/self/exe", ...)) and under which folder it should create the DLS subfolder to store user-generated content. It first looks for the HOME environment variable. Then for the XDG_DATA_HOME environment variable and if none is defined, it uses the installation path.
After that the log is opened at <user-data-folder>/Documents/DLS/DLS.log
If you cannot find the log file, it means that the first step failed. Which either means that the above procedure has a bug (which I haven't found yet), or the paths are correctly read but DLS doesn't have permissions to create files and subfolders in them.
Will try to debug it by trying out all the different cases in the above procedure and will report back.
Thanks for trying to figure it out and sorry again for the trouble.