tosyokan no pasokon wo tukatta ato ha, kanarazu roguautosuru you ni site imasu.

図書館のパソコン means the library's computers or the computers at the library.

Here, の links two nouns:

• 図書館 = library
• パソコン = computer

So 図書館のパソコン means the computers that belong to, or are associated with, the library.

If you said 図書館でパソコン, that would mean something more like at the library, a computer..., where で marks the place where an action happens. That is a different structure.

In this sentence, the speaker is not just talking about using a computer in the library in general, but specifically using the library's computer(s).

を marks the direct object of the verb.

The verb here is 使う = to use, so the thing being used is marked by を:

• パソコンを使う = to use a computer

So in:

• 図書館のパソコンを使ったあと

the meaning is:

• after using the library computer(s)

When あと means after doing..., Japanese normally uses the plain past form of the verb before it:

• 食べたあと = after eating
• 見たあと = after seeing
• 使ったあと = after using

So:

• 使ったあと literally means the after of having used

This is a very common pattern:

• Verb (past plain) + あとで / あと
• Noun + の + あとで / あと

So 使ったあと is the natural form here.

In 使ったあとは, the は marks the time/occasion after using it as the topic or point of focus.

Compare:

• 使ったあと、ログアウトします
  = After using it, I log out.

• 使ったあとは、ログアウトします
  = As for after using it, I log out. / After using it, I make sure to log out.

The は adds a slight sense of contrast or emphasis. It helps highlight what one does after using the computer. In natural English, this often just sounds like emphasis rather than a strong contrast.

必ず means certainly, without fail, or always in the sense of making sure it happens.

So:

• 必ずログアウトする = make sure to log out

It adds a sense of rule, habit, or obligation:

• not just I log out
• but I always make sure to log out

This fits the context well, since logging out of a public computer is a safety/security habit.

ログアウト is a loanword noun from English log out. In Japanese, many loanwords become verbal expressions by adding する.

So:

• ログアウトする = to log out
• コピーする = to copy
• チェックする = to check

This is very common in Japanese. Even if the original English word is a verb, Japanese often treats it as a noun-like word and adds する to make it into a verb.

ようにしている is a very important pattern. It means something like:

• to make a point of doing
• to try to do regularly
• to keep doing as a habit
• to make it a rule to do

So:

• ログアウトするようにしている means:
• I make a habit of logging out
• I make sure to log out
• I try to always log out

This is different from simply saying:

• ログアウトする = I log out

The ようにしている part emphasizes ongoing practice or conscious habit.

ようにする means to make an effort to do / to make it a rule to do.

When changed to ようにしている, it shows that this effort or rule is ongoing:

• ようにする = decide to do / try to do
• ようにしている = be in the habit of doing / keep making an effort to do

So:

• ログアウトするようにする
  = I will try to log out / I make it a rule to log out

• ログアウトするようにしている
  = I am making it a habit to log out / I usually make sure to log out

The sentence uses しています because it describes a current habit or regular practice.

The sentence is written in a polite style because the main verb is polite:

• しています instead of している

This matches many everyday written and spoken contexts, especially when explaining one's habits politely.

Compare:

• ログアウトするようにしている
  plain / casual

• ログアウトするようにしています
  polite

Everything before that can stay in plain form, because Japanese often uses plain forms inside clauses even when the whole sentence ends politely.

For example:

• 使った is plain past
• しています is polite present progressive

This mix is completely normal.

It can be either, depending on context.

Japanese nouns usually do not mark singular vs. plural unless necessary. So:

• 図書館のパソコン can mean
  • the library computer
  • the library computers
  • a computer at the library

In this sentence, English might naturally translate it as:

• After using a library computer... or
• After using the library computers...

The exact number is not the main point. The habit of logging out is what matters.

The subject is omitted, which is very common in Japanese.

From しています, the natural interpretation is:

• I am making sure to log out or possibly
• we make sure to log out

In most everyday contexts, English would supply I:

• After using the library computer, I always make sure to log out.

Japanese often leaves out subjects when they are clear from context.

Yes, in many cases あと and あとで are both possible, though the nuance can be slightly different.

• 使ったあと = after using
• 使ったあとで = after using / after having used

In this sentence, both would sound natural:

• 図書館のパソコンを使ったあとは、...
• 図書館のパソコンを使ったあとで、...

The version with は in あとは slightly emphasizes what happens after use. That is probably why this form was chosen.

It could, but the nuance would be different.

For example:

• 図書館のパソコンを使ったら、必ずログアウトするようにしています。

This is understandable, but たら often sounds more like when/if you use it, then... or once you've used it... It connects events, but it does not focus as directly on the time point after doing something as あと does.

使ったあと is especially natural when talking about a routine action performed after another action:

• after eating, brush your teeth
• after using the computer, log out

So あと is the better fit here.

Yes. パソコン usually means personal computer and is very commonly used in everyday Japanese for desktop or laptop computers.

コンピューター is also correct, but it can sound broader or slightly less everyday depending on context.

In places like libraries, schools, or offices, パソコン is a very natural word for the machines people use.

So:

• 図書館のパソコン sounds very normal and practical.

This is a very common feature of Japanese.

Only the final predicate of the main sentence needs to show the overall politeness level. Earlier verbs inside modifying clauses often stay in plain form.

So in this sentence:

• 使った is plain past, because it modifies あと
• しています is polite, because it ends the sentence

This is normal Japanese grammar, not a mismatch.

You can think of it like this:

• subordinate parts: often plain
• sentence ending: shows politeness

So the sentence is fully polite even though 使った is not polite form.