How long does it take before Google finds a backlink?

Basically the question I am asking is how long does a link have to stay up before Google will index it and validate it as a backlink. Would the link have to stay up until the next Google Pagerank update for me to see a change in my pagerank?

Google searches pages that are updated on a regular basis, there is no point Google going to back to a static page that is not updated (probably only once a year update sort of thing) because it is a waste of its time for the use of Google's resources. So it indexes the sites that are updated regularly and then searches them on a x time period. The PR ranking system as far as I know, is not updated daily (could be wrong here ?). So it may take a couple of days to go up in the Google page rank once the backlink has been noticed.

Ideally you would want backlinks to remain indefinitely.

How do you search for your backlinks?

Both Google and Yahoo have their own methods of showing you where other sites are linking to you that they know about. Google via Webmaster Tools and Yahoo via Site Explorer.

What is the difference between Yahoo's method and Google's method then? I am always curious about this because one of my blogs rank pretty good on Google and suck on Yahoo.

They have a different algorithm and ways of showing backlinks.