Log space consumed in creating index

If the recovery mode is in full, it will log the whole table(data + indexes) in the transaction log file, even if the new index is a non-clustered index. That is surprised to me. I think it will only log the new index.
For example, we have the customer table(30GB data), it has three indexes(10GB). For the following example:
Create index ix_date on customer (create_date) with (online = on).
The size of new index is 3GB. The transaction log file will use 43GB.