I stumbled in here looking for a similar answer without the "git log" restriction. The answers here didn't give me what I needed but this did so I'll add it in case others find it useful:
git diff --name-only
You can also couple this with standard commit pointers to see what has changed since a particular commit:
git diff --name-only HEAD~3
git diff --name-only develop
git diff --name-only 5890e37..ebbf4c0
This succinctly provides file names only which is great for scripting. For example:
git diff --name-only develop | while read changed_file; do echo "This changed from the develop version: $changed_file"; done
#OR
git diff --name-only develop | xargs tar cvf changes.tar