When i boot my xbox up the tv will stay blank but i will hear the sounds that it usaully makes when it logs into my account.

SOURCE: My xbox 360 wont show a picture but it has sound

when we experience a no video/display but with sound as if your logging in at your default blade, try to disconnect all the cables including the power cord and the audio video cable. (its the cable going from your console or xbox 360 to your tv). Before you plug it back try also to wipe the prongs or the metal part of the cables to ensure it has no debris or anything that would mess up the connection. If your using an HDTV make sure that the switch of your component cable is click on the hdtv settings. if it still not working for you, try to use a different cable and TV just to isolate the problem. If the issue still persist.. its the system that has a problem. Not exactly the hardware but it would definitely the prongs at the back of the console. So, bring it back to the retailer if its still in 30-days warranty or send for repair to its authorized service center.

SOURCE: xbox 360

Check your AV cable if your tv is HD push the tab to HD if not set it to normal tv this should help you out. If that doesnt work i would try to borrow someones AV cable and try that.

SOURCE: My xbox 360 wont show a picture but it has sound

Since your sound works, you know the xbox is booting properly.  Check two things: 1) try another television to make sure it's not the tv.  2) try another cable(from a friend?) to see if the cable is bad.
**Also, if the video settings were saved on an unsupported setting, it could cause the screen to not show video.  If you try these 2 things and you think someone might have saved bad settings, I recall a button combination you can use to reset certain things, and I try to help if it comes to that.

SOURCE: When i boot my xbox up the tv will stay blank but

If your Xbox 360 does not have display you need to stop the heat getting to the chips. You can actually do this yourself as many have in this forum using the recommended Xbox 360 Repair guide.

Basically the guide will show you how to stop the heat build up with videos and household items, of course if you are in doubt send it in.

It has a preview video on the website.

Or of course you can send it in to have it fixed.

SOURCE: When i boot my xbox up the tv will stay blank but

This Problem is well known threw out most 360 repair shops.
The G.P.U. has lost the video feed due to thermal expansion
of the P.C. board. The remedy 9/10 is to have the G.P.U
re-flowed and then securely clamp to the heat sink. This
will yield the most durable repair . Cost will very 60.00-80.00
depending on warranty and track record of the shop.

If you need more info feel free to ask.

Rob