It’s easy. Just read The Letter of Aristeas, as well as Philo and Josephus (Antiquities 12.1-118), some of our earliest sources for the legend of the seventy-two translators. They keep referring to the work being translated as “the Law.” The Jews actually kept the correct identification; it’s just us Christians who have extended the term.
For it was taught: King Ptolemy gathered seventy-two elders, set them in seventy-two houses and did not disclose why he had put them there. He visited each one [separately] and said to him, ‘Write out for me the Torah of your teacher, Moses!’ The Holy One, blessed be He, put counsel into the mind of each one, so that they were all of one opinion.
- Babylonian Talmud, Tractate Megillah 9a-9b
Most serious biblical scholars attest that St. Paul references the LXX as the Septuagint is called in several of his epistles as well as Jesus himself.
This pretty much sums it nicely.
I]t is necessary to clarify what we mean by “the Septuagint.” Sometime between three to two hundred years before Jesus and Hillel, Greek-speaking Jews began producing translations of books that we now find in the Hebrew Bible: they started with the Pentateuch, the first five books of the Bible, but eventually translated them all. There is a legend that comes to us in a fictional work called the Letter of Aristeas. It tells how seventy-two Jewish men were gathered to the island of Pharos in Alexandria by an Egyptian ruler in the mid-third century BCE to translate the first five books of the Bible. Over time, the number seventy-two became remembered as seventy for some reason, and in Latin seventy is septuaginta; that name stuck with the earliest Greek translation of various books, and not only the first five books, but all the books classed together as authoritative Jewish Scriptures.
Today Septuagint is a fuzzy term. Some people think of a particular collection as being “The Septuagint.” Due to its widespread use today, students, pastors, and scholars think of the German textual expert Alfred Rahlf’s Septuaginta. It is available in a handy one-volume print edition, and since at least the 1980s its text has been available in electronic format. Most scholars do not actually believe that there was a book (codex) form of the Septuagint at the turn of the millennia, but they still talk about “the Septuagint,” as if Paul, for example, had gone to the local Alexandrian Bible Society bookstore in Tarsus and purchased his own bound copy of The Septuagint, from which he preached. In more thoughtful moments, we realize that Paul would not have done that, but rather he cited the Septuagint probably from a combination of memory, some form of crib sheet with important texts written out, and by finding and quoting from locally available scrolls or copies of particular biblical books that belonged to the translation-tradition that we refer to as “the Septuagint.” Some scholars will use the term Septuagint more loosely to refer to any Greek version of the Scriptures that were available to the early church. Still others reserve “Septuagint” for the earliest translation of the first five books of the Bible, and they use “Old Greek” for the first translations of the other books.