tosyokan no pasokon wo tukaiowattara, roguautosite kudasai.

A natural reading is:

としょかん の パソコン を つかいおわったら、ログアウトして ください。

Romanization:

Toshokan no pasokon o tsukaiowattara, roguauto shite kudasai.

Notes:

• を is pronounced o
• 図書館 = としょかん
• 使い終わったら is a long form, so it helps to hear the parts:
  • 使い = use
  • 終わったら = when/after finishing

Here, の links two nouns and shows possession or association.

So:

• 図書館 = library
• 図書館のパソコン = the library's computers / computers in the library

In English, we might say:

• the library computers
• the computers at the library
• the library's computers

Japanese uses の very often for this kind of noun-to-noun connection.

を marks the direct object of the verb.

In this sentence, the thing being used is パソコン, so it takes を:

• パソコンを使う = to use a computer

So the first part means:

• 図書館のパソコンを使い終わったら = when/after you finish using the library computer

使い終わったら means when you finish using or after you have finished using.

It is built from:

• 使う = to use
• 使い終わる = to finish using
• 使い終わった = finished using
• 使い終わったら = when/after finishing using

So 〜終わる added to the stem of a verb often means finish doing that action:

• 読み終わる = finish reading
• 食べ終わる = finish eating
• 使い終わる = finish using

This makes the sentence more specific than just 使ったら, which would mean when/after you use it.
使い終わったら clearly means once you are done using it.

〜たら can mean different things depending on context, including if, when, or after.

In this sentence, it means something like:

• when you finish using it
• after you finish using it

That is because this is a real, expected sequence of actions:

1. Use the library computer
2. Finish using it
3. Log out

So here 〜たら is not really a hypothetical if. It is more like once you have done X, do Y.

Japanese often combines the ます-stem of one verb with another verb to create a compound verb.

So:

• 使う → 使い (stem)
• 終わる = finish
• 使い終わる = finish using

This is a very common pattern:

• 書き始める = begin writing
• 読み続ける = continue reading
• 食べ終わる = finish eating

You could sometimes express similar ideas in other ways, but 使い終わる is a standard, natural compound verb.

〜てください is a polite way to tell someone to do something.

So:

• ログアウトする = to log out
• ログアウトして = log out (te-form)
• ログアウトしてください = please log out

This is very common in instructions, signs, announcements, and polite requests.

The whole sentence is basically an instruction:

• After you finish using the library computer, please log out.

Grammatically, 〜てください is a polite request.
But in public notices and rules, it often functions like a polite instruction or command.

So on a sign like this, it means something close to:

• Please log out
• Be sure to log out
• Log out, please

It is polite, but it still clearly tells the reader what they are expected to do.

ログアウト is a loanword from English log out, so it is written in katakana, which is commonly used for:

• foreign loanwords
• foreign names
• some technical terms
• emphasis in some cases

Japanese computer-related vocabulary often uses katakana:

• パソコン = personal computer
• ログイン = log in
• ログアウト = log out
• クリック = click

パソコン is short for パーソナルコンピューター.

It usually means:

• personal computer
• PC

In many everyday situations, Japanese speakers say パソコン rather than a longer word like コンピューター.

So 図書館のパソコン most naturally means:

• the library computers
• the computers provided by the library

Japanese often leaves out the subject when it is obvious from context.

In a notice or instruction like this, the implied subject is understood to be:

• you
• the person using the computer

So even though Japanese does not say you, the meaning is still clear:

• When you finish using the library computer, please log out.

This omission is extremely common in Japanese.

The comma helps separate the condition/time clause from the main instruction.

So the sentence is divided like this:

• 図書館のパソコンを使い終わったら、
• ログアウトしてください。

That makes it easier to read:

• After you finish using the library computer,
• please log out.

Japanese commas often mark a pause or show the structure of the sentence more clearly.

Yes, that is a very natural translation.

Other good translations include:

• After using the library computer, please log out.
• When you are finished using the library computer, please log out.
• Please log out after using the library computer.

Even if the exact English wording changes, the core Japanese grammar stays the same.

It is polite and appropriate for a public notice.

Why:

• してください is polite
• The sentence is direct, but not rude
• It sounds like standard institutional Japanese, suitable for a library, school, or office

It is not casual conversation, but it is also not extremely stiff or ceremonial. It is the kind of polite wording you would expect on a sign or instruction notice.