Do they have slight difference in meaning when I use "as" and "since" as synonym of "because" ?
e.g.
I had to shut down the computer since it got infected by virus.
I had to shut down the computer as it got infected by virus.

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Answer

"As" and "since" can be used as synonyms of because. The words do have other definitions besides that one, but your understanding is correct.

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In American English, "since" and "as" can  both be used as synonyms of  "because."  However, in conversation, ¨because"  or it´s abbreviated form "´cause"   would be much more commonly heard.   "Since" and "as" in the sentence you gave sound a bit formal or bookish. They sound better in writing than in conversation.  In British English they might be more conversational.

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since -- time
as -- reason

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