Logo depois da prova, eu vou ligar para minha mãe.

Here, logo means right or immediately. So logo depois da prova means right after the exam/test, not just after the exam/test.

• depois da prova = after the test
• logo depois da prova = right after the test / immediately after the test

In other contexts, logo can also mean soon, so it is a useful word to notice.

Because de + a contracts to da in Portuguese.

• de = of / from / after, depending on context
• a prova = the test

So:

• de a prova → da prova

This kind of contraction is very common:

• de + o = do
• de + a = da
• de + os = dos
• de + as = das

In depois da prova, the underlying structure is depois de a prova, but Portuguese normally contracts it to depois da prova.

Yes, when depois is followed by a noun or pronoun, it normally uses de.

Examples:

• depois da aula = after class
• depois do trabalho = after work
• depois de mim = after me

So in your sentence:

• depois da prova = after the test

If a verb comes after it, you also use de:

• depois de estudar = after studying
• depois de fazer a prova = after taking the test

In Brazilian Portuguese, prova usually means a school test or exam, especially a more formal one.

Roughly speaking:

• prova = test/exam, especially in school or college
• teste = test, but often something shorter, more informal, or a trial/test run
• exame = exam, often more formal; it can also mean a medical exam

So in a school context, prova is extremely common and natural.

Because ir + infinitive is one of the most common ways to talk about the future in Brazilian Portuguese.

• eu vou ligar = I’m going to call / I will call
• eu ligarei = I will call

Both are correct, but eu vou ligar sounds much more natural in everyday spoken Brazilian Portuguese.
Ligarei is grammatical, but it can sound more formal, written, or less conversational.

No, it is not strictly necessary.

You can say:

• Logo depois da prova, eu vou ligar para minha mãe.
• Logo depois da prova, vou ligar para minha mãe.

Both are correct.

Portuguese often allows the subject pronoun to be omitted when the verb already makes the subject clear. Here vou already tells you it is I.

That said, in Brazilian Portuguese, speakers often include subject pronouns more than in some other varieties of Portuguese, so eu sounds perfectly natural.

Because when ligar means to call someone on the phone, Brazilian Portuguese usually uses para (or the informal pra).

So:

• ligar para alguém = to call someone

Examples:

• Vou ligar para ele. = I’m going to call him.
• Ela ligou para a amiga. = She called her friend.

This is different from English, where you usually say call someone without a preposition.

Also, be careful: ligar can mean different things depending on context:

• ligar para alguém = to call someone
• ligar a TV = to turn on the TV

Yes. In spoken Brazilian Portuguese, pra is extremely common.

So these both work:

• vou ligar para minha mãe
• vou ligar pra minha mãe

The version with para is a bit more neutral or formal in writing.
The version with pra is very natural in conversation.

Because minha has to agree with mãe, and mãe is feminine.

Possessive words agree with the thing possessed, not with the speaker.

So:

• meu pai = my father
• minha mãe = my mother
• meu livro = my book
• minha casa = my house

Since mãe is feminine singular, the correct form is minha.

Both are possible in Portuguese, depending on region, style, and context.

• minha mãe
• a minha mãe

In Brazilian Portuguese, minha mãe without the article is very common and completely natural, especially with close family terms.

Using the article can also be correct, but it may sound a little more regional, emphatic, or stylistically different depending on the speaker.

So for a learner, minha mãe is a very safe and natural choice.

It is very common and natural, because Logo depois da prova is an introductory time expression.

• Logo depois da prova, eu vou ligar para minha mãe.

The comma helps separate the time phrase from the main clause.

In informal writing, some people might leave it out:

• Logo depois da prova eu vou ligar para minha mãe.

That is still understandable, but the comma is a good choice and often preferred.

Mãe can be tricky for English speakers because it has a nasal sound.

A simple guide:

• m as in English mother
• ãe is a nasal diphthong, something like my but nasalized

It is not exactly like any normal English vowel. The sound comes through the nose more.

A rough approximation is:

• mãe ≈ myng without a full ng sound

But that is only approximate. The important thing is that the vowel is nasal.

Also note the tilde:

• ã signals nasalization

So mãe is different from a non-nasal vowel sound.