Learn Portuguese (Portugal): Até logo, mãe!

The word até means "until" and logo means "later" or "soon". So, the phrase literally translates to "until later". It is one of the most common ways to say goodbye in Portuguese.

When exactly should I use Até logo instead of Adeus?

You use Até logo when you expect to see the person again later the exact same day. If you are not going to see them again today, you would usually say Adeus (goodbye) or Até amanhã (see you tomorrow).

How do I pronounce the wavy line over the letter a in mãe?

That wavy line is called a tilde and it tells you to make a nasal sound. You pronounce mãe a bit like "my" in English, but you push the sound through your nose, similar to the hum at the end of the English word "sing".

Why isn't there a word like "the" or "my" before mãe?

Because you are addressing your mum directly to say goodbye, you just use the word mãe by itself as her name, exactly like you would in English!