The logic is given by Rambam in Mishneh Torah Melakhim uMilchamot 9:1:
על ששה דברים נצטווה אדם הראשון [...] הוסיף לנח אבר מן החי שנאמר אך בשר בנפשו דמו לא תאכלו נמצאו שבע מצות
Six precepts were commanded to Adam. [...] The prohibition against eating flesh from a living animal was added for Noah, as it is stated (Bereishit 9:4): 'Nevertheless, you may not eat flesh with its life, which is its blood.' Thus there are seven mitzvot.
(See also Sanhedrin 56a and Bereishit Rabbah 34:8)
So you have to go back to Adam to see these six laws that are listed with their sources in Bereishit Rabbah 16:6. Five are based on the method called gezeirah shavah that derives teachings based on common words in different verses. Bereishit 2:16 was divided into six parts by the sages referring to six commandments:
- idolatry – וַיְצַו: based on the common word צו also found in Hoshea 5:11
- blasphemy – 'ה: Hashem alludes to Vayikra 24:16
- judges – אֱלֹקִים: the other divine name, Elokim alludes to Shemot 22:27
- (bloodshed – עַל־הָאָדָם: man alludes to Bereishit 9:6)
- incest – לֵאמֹר: saying alludes to Yirmiyahu 3:1
- theft – מִכֹּל עֵץ־הַגָּן אָכֹל תֹּאכֵל: Of every tree of the garden thou mayest freely eat alludes to the fact that one needs a permission from an owner to take her/his property.
(See also Sanhedrin 56b)
Another great article from MyJewishLearning.com regarding the question.