You can make a tar anywhere but think about what you are telling tar to do...
If you tell tar to backup /root/Documents/daily-logs/24dec its going to backup just that, the entire path and meta data for the files. This is by the design of tar, imagine for a moment that you have developed an application with a big directory structure that you want to be able to port to another system. This app is contained in /opt/app1 with multiple folders underneath, /opt/app1/folder1, /opt/app1/folder2, /opt/app1/folder1/subfolder1, /opt/app1/folder1/subfolder1/subfolder2.
You are currently in the root home directory, /root.
You tell tar to make a backup with the following:
tar -czvf /tmp/myapp1.tar.gz /opt/app1
Tar will backup the files and folders as:
So when you go to deploy the tarball on another system the directory structure is the same as it was on the original system.
Note that tar will be default remove the leading / from file and folder paths.