Is the word lógico the same in Brazilian and European Portuguese?

Yes.

The word lógico is identical in meaning, spelling, and grammatical function in both Brazilian and European Portuguese. It can be used as an adjective (meaning "logical" or "rational") or as an interjection (meaning "of course" or "obviously"). The only difference is in pronunciation: in Brazilian Portuguese, the unstressed vowels tend to be more open, whereas in European Portuguese, unstressed vowels are often more closed or even reduced to the point of being almost silent.

Brazilian Portuguese Examples

1. É lógico que ele vai vir à festa. (It's logical/obvious that he is going to come to the party.)
2. Lógico! Eu te ajudo com essa mudança. (Of course! I'll help you with this move.)
3. Esse argumento não parece muito lógico para mim. (This argument doesn't seem very logical to me.)
4. É lógico que você esteja cansado depois de trabalhar tanto. (It's logical that you are tired after working so much.)
5. Seguir um plano lógico é o melhor caminho. (Following a logical plan is the best path.)

European Portuguese Examples

1. É lógico que ele venha à festa. (It's logical/obvious that he comes to the party.)
2. Lógico! Eu ajudo-te com essa mudança. (Of course! I'll help you with this move.)
3. Esse argumento não me parece muito lógico. (This argument doesn't seem very logical to me.)
4. É lógico que estejas cansado depois de trabalhar tanto. (It's logical that you are tired after working so much.)
5. Seguir um plano lógico é o melhor caminho. (Following a logical plan is the best path.)