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.

Question

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.

Answer

since -- time
as -- reason