Here's my attempt of an answer (I might have to post it unfinished and come back to it later). I assume that your native language is English.
Take as well as Russian взять / брать are the examples of highly polisemous verbs.
From here:
This paper attempts to show what the definition of a word should look
like, in order to be of use to AI programs such as analyzers and
generators. We concentrate on take, a highly polysemous word which
poses many vexing questions. Why is it that we can say take a
look or take a whiff, but not take a stare or take a smell? Why can a
photographer take a picture, while an artist cannot take a painting?
It is clear that there must be several distinct senses of take,
because the implications are different for different usages. For
example, when I take a book (or when I am given a book) it is
felicitous to say I got a book or I have a book. However, if I take it
in stride, it is not the case that I got it in stride or I have it in
stride. Faced with these facts about language, our goal is to come up
with definitions for the various senses of a word...
And the sets of their meanings/senses in English and Russian overlap only partially.
Basically, this is the reason why взять производную is ok while взять экзамен is not.
Even more interestingly - взять такси is perfectly fine while взять автобус/поезд is something you can mostly hear from emigrants (and non-native speakers, of course) who sometimes use calques of the corresponding English/French expressions.
Now to your list of examples. These days one can use Google as a very convenient tool to look up the frequency of occurrences of a particular expression. This normally gives a good indication of whether a phrase is well-established or not.
You could easily do this yourself, so I'll be brief:
Does взять work generally as a synonym for вычислить or применить операцию:
No.
прологарифмировать числo = взять логарифм числа?
Yes (or rather sounds good enough to me, professional mathematicians may disagree).
возвести число в квадрат = взять квадрат числа?
No. взять sounds weird.
возвести обе стороны уравнения в степень = взять степень обеих сторон уравнения?
No. взять sounds weird.
извлечь квадратный корень числа = взять квадратный корень числа?
Hmm... Sounds a bit odd but I'm not sure I'd notice this oddity if I didn't focus on this specifically (as I'm doing now)...
проинтегрировать функцию = взять интеграл функции?
Yes.
вычислить когомологии короткой точной последовательности = взять когомологии короткой точной последовательности?
You've lost me here :) But intuitively, w/out googling - No. найти / определить maybe but not взять.
