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I accidentally pulled the connector off. So my macbook air only works when its plugged it. How does one get this repair without having to replace the logic board. Where does it need to be solder at? Any help would greatly be appreciated.

Which connector are you taking about? If you’re taking about the battery connector, you will most likely have to get one off a donor board, and you will need a soldering iron and solder sucker/wick as it is soldered as a through-hole component. If this is another connector that you need to replace, then you might need a more specialized soldering iron and potentially hot air station as well to remove and re-solder as the rest of the connectors to the motherboard are surface-mounted. Most of the other connectors are available from various mac parts retailers online.