Is there any contextual difference between 'as far as I know' and 'as far as I can ascertain'?

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    You can often ascertain the answer to something you don't know by googling it. – Ronald Sole Dec 5 '16 at 0:32

The difference is in the implied expenditure of effort.

"As far as I know" implies simply recalling from memory, whereas "as far as I can ascertain" implies that one has actually done some research on the question (setting aside the actual amount or the truthfulness of the material he/she pulled up).

Then again, depending on how precise the speaker is with their language, there may not be an actual difference (though anyone who uses "ascertain" is, in my opinion, more likely to actually look stuff up).

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