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Después vs. luego

While both después and luego can often be translated as "then" or "afterwards", they are not always interchangeable. The main difference lies in the implied time gap between events and whether there is a logical connection between them.

Después

A1
This is the most common and general word for "after", "afterwards", or "later". It indicates that one event follows another, without specifying how much time has passed. It can refer to a gap of seconds, minutes, days, or even years.
Primero vamos a la playa y después a comer.
(First we are going to the beach and afterwards to eat.)
Llámame después de la reunión, por favor.
(Call me after the meeting, please.)
Se casaron y tuvieron hijos tres años después.
(They got married and had children three years later.)
¿Qué quieres hacer después?
(What do you want to do afterwards?)
Nos vemos después en el café.
(We'll see each other later at the café.)

Luego

A2
This word also means "then" or "next", but it often implies a more immediate succession of events than después. It can also be used to indicate a logical consequence, much like "so" or "therefore".
Termina tu sopa y luego comes el postre.
(Finish your soup and then you eat dessert.)
Pienso, luego existo.
(I think, therefore I am.)
Gira en la primera calle a la derecha y luego sigue recto.
(Turn at the first street on the right and then continue straight.)
Si no estudias, luego no te quejes de tus notas.
(If you don't study, then don't complain about your grades.)
Bueno, me voy ya. ¡Hasta luego!
(Well, I'm leaving now. See you later!)

Summary

In short, use después as a general-purpose word for "after" or "later" for any time frame. Use luego when you want to emphasize an immediate sequence of actions ("right after") or to show a logical result or consequence ("so" or "therefore"). While in many daily situations they can be used interchangeably, luego adds a sense of immediacy or causality that después lacks.