[noun] + に基づいて [noun] + に基づき [noun] + に基づいた + [noun] [noun] + に基づく + [noun]
に基づいて means "based on", "in accordance with": the noun before it – facts, data, a law, a contract – is the ground that an action or a judgment stands on. It goes straight after a noun.
The basis takes に, because もとづく基づく is an intransitive verb: one thing rests on another. It never takes を.
Before a noun, the pattern becomes にもとづいた基づいた or にもとづく基づく. The two mean the same; に基づく sounds a little more formal.
に基づいて belongs to formal speech and writing: law, news, reports, business. In writing the て drops: にもとづき基づき.