Log [in into/into/onto] your account/laptop/the website.

Which are correct? Could you also explain if these prepositions actually have different meanings?

Thanks in advance
As most native speakers of English are not that computer literate you'll hear various 'Logisms', in fact Wikipedia quotes the following:

To log in (also: to log on, sign in, or sign on) is to identify oneself to the system in order to obtain access.

To log out (also: to log off, sign out, or sign off) is to close off one's access to a computer system after previously logging in.


Personally I would use them in the following way:

Log into (Login) an account or a website account (if you're a member)

Log onto (Logon) a network

Of course as most computers are networked in offices and now that many people have home networks, then you'll log into your account and log onto the network simultaneously.

However, as there is no clear definition or acceptance of the difference by most native speakers, don't be surprised if you hear any or all of the above being used interchangeably.
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The difference between 'in to' and 'into' or 'on to' and 'on to' really has to do with inmplied motion/direction:

Jack went in to say, "Hello" to an old friend. (The to is part of the infinitive, not relating to the 'went in' part.)
Jack went into the room. (Motion)

He threw the keys onto the table. (Here we have motion of keys onto table)
He logged on to see the news. (Here again, the 'to' relates to the verb 'see' rather than to 'logged on';
He logged onto the computer. (Implied motion: moving 'into' the computer).

In any case, I've found that this way of viewing things correlates well with Slavic language verbs of motion. I hope it helps!
I disagree with Tam and Christopher. "To log on/in" is a verb phrase. The helper word "on/in" in the verb phrase should not be concatenated with the preposition "to" in either "to the network" or "to your account." If someone asks you "Did you log on to your account?" You don't reply, "Yes, I logged." You say, "Yes, I logged on."

This simplifies the rule: Log on/log in for verbs. Logon/login for nouns and adjectives.

This also helps for initial caps rules: Log On with Your User ID and Password
Log On here is the verb phrase; therefore, "on" takes an initial caps.
"...most native speakers of English are not that computer literate..."

Just because you start your claim to knowledge with an insult, doesn't mean you're an authority to be taken seriously.
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Microsoft Style Guide (4th edition) recommends to "use log on or log on to to refer to creating a user session on a computer or a network. Use log off or log off from to refer to ending a user session on a computer or a network. Use sign in and sign out to refer to creating and ending a user session on the Internet. Do not use log in, login, log onto, log off of, log out, logout, sign off, or sign on..."

Logged onto the site