We have log backups running every 15 minutes and nightly fulls/diffs (fulls are once a week per DB, but they are spread across the whole week for load balancing). Lets say for a particular DB I have a diff that starts at 10:05pm and finishes at 10:18pm and I have log backups at 10:00pm, 10:15pm, and 10:30pm. If I want to restore to 10:24pm that night, I know I need to restore the last full and the diff from that night, but then I'm not sure which transaction log I should start with after that.
First thing if transaction log is not damaged you should go for tail log backup. That would cause minimal data loss. Then restore would be
Weekly full backup with no recovery
10:05PM Differential backup with no recovery
All log backups after this differential backup with no recovery and finally tail log backup with stopat command with recovery
Please read about stopat command
Make sure you don't forget to test backup validity before restoring just in case.