At what age may a student log time and use that time to obtain a rating?
FAR § 61.83 states:

To be eligible for a student pilot certificate, an applicant must:
(a) Be at least 16 years of age for other than the operation of a glider or balloon.
(b) Be at least 14 years of age for the operation of a glider or balloon.
(c) Be able to read, speak, write, and understand the English language. [...]

A student pilot certificate enables you to fly a plane solo or on a solo cross country.
In other words, you need to be at least 16 years old to fly a plane (non glider/balloon) solo.

Now, back to your initial question. If you go through FAR §61.83 - §61.95 you will find all the regulations about student pilots (which would be the kind of pilot you are referring to ).
I have found no age restriction going through them for the pre-solo instruction. All it says that it must be done before soloing which does require a medical exam and certificate (but these don't have age restrictions either).
Summing up, a 9 or 12 year old child is legally able to take flying lessons with an instructor, but may not solo before his/her 16th birthday (no glider/balloons).