When you backup the transaction log, the backup includes all events that has occurred since the most recent log backup (with some qualifications, see below). Whether has been a full backup in between is irrelevant.
When you restore a full backup + a tranlog backup, the tranlog backup may start before the full backup started and end after the full backup ended. That is not a problem. SQL Server knows where to start reading in the log backup, and ignore the records that are in the full backup.
I don't know what the exact requirements are of your client are. Doing like you do above is not overly useful, since you take the transaction log backup directly after the full backup. There will not be much use of it.
But maybe your client wants you to send an initial full backup, and then send them transaction log backups daily, which will be smaller and take less time to apply. This is perfectly possibly to do, provided that they restore the database with the STANDBY option, which leaves it a readonly state.
If you don't know, I think you should ask your client for more details on this requirement.
As I said "the backup includes all events that has occurred since the most recent log backup with some qualifications". Let's look in to these qualifications:
- The database has always been in full recovery. If you switch to simple recovery, you break the log chain. This is very important to observe!
- There are some options which does not truncate the transaction log, which means that the next transaction log will include everything that is in the log backup you are currently taking.