Log in to your account is correct. To log in is a phrasal verb.
The phrasal verb “To log in” can take different prepositions and to is just one example.
Log in at the desk.
Log in over there.
You need to log in by 9 pm.
We do not combine in and at and neither do we combine In and to in this context.
You need to log in to your account.
You need to log into your account.
When to use into?
Into often has the meaning of transformation.
He turned into a monster.
He turned in to the gas station.
We also use into when we combine the prepositions in and to(but NOT when in is part of a phrasal verb).
Let’s move into the sitting room.
In this case, we need the preposition “in” to show the space of the sitting room and “to” for movement. This is the situation when you can combine “in” and “to” to form into.
Log-in as a noun
Phrasal verbs can also be used as nouns(especially in spoken English).
Where is the log-in sheet?
What is your log-in number?
In more informal English, it is common to drop the hyphen.
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