Ttdfjt

Recently I found out from reading a text that "взять производную" (literally the same as "take the derivative" in English) is an acceptable synonym for продифференцировать, which surprised me since I'd heard that the literal translation взять экзамен from English is incorrect (makes the person sound like a thief).

Does взять work generally as a synonym for вычислить or применить операцию:

прологарифмировать числo = взять логарифм числа?

возвести число в квадрат = взять квадрат числа?

возвести обе стороны уравнения в степень = взять степень обеих сторон уравнения?

извлечь квадратный корень числа = взять квадратный корень числа?

проинтегрировать функцию = взять интеграл функции?

вычислить когомологии короткой точной последовательности = взять когомологии короткой точной последовательности?

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.

In a casual or semi-casual conversation about math взять can work as a synonym for вычислить. I can easily imagine a math lecture containing phases like

Теперь давайте возьмем от этого выражения логарифм.

Берите от функции квадрат, чтобы в задаче на оптимизацию был подходящий градиент.

Same for взять квадрат, интеграл, сумму, корень, когомологию. Note however that this is not the most formal way to talk, I wouldn't do it in a publication.