turns off on startup after update

Did you try turning the computer back on? Also try holding on the power button for about 5 seconds. Hope that helps, I never really had that issue.

You need a minimum of about 15 GBs to do an installation of OS X. About 10 GBs for a basic update. Be sure you have adequate free space. You need 20 GBs or 10% of the hard drive's capacity as free space, whichever is greater.

Freeing Up Space on The Hard Drive

1. See Lion/Mountain Lion/Mavericks' Storage Display.

2. You can remove data from your Home folder except for the /Home/Library/ folder.

3. Visit The XLab FAQs and read the FAQ on freeing up space on your hard drive.

4. Also see Freeing space on your Mac OS X startup disk.

5. See Where did my Disk Space go?.

6. See The Storage Display.

You must Empty the Trash in order to recover the space they occupied on the hard drive.

You should consider replacing the drive with a larger one. Check out OWC for drives, tutorials, and toolkits.

Try using OmniDiskSweeper 1.8 or GrandPerspective to search your drive for large files and where they are located.

This 'might' help recover your Mac: OS X: About OS X Recovery - Apple Support

Reinstalling OS X Without Erasing the Drive

Boot to the Recovery HD: Restart the computer and after the chime press and hold down the COMMAND and R keys until the menu screen appears. Alternatively, restart the computer and after the chime press and hold down the OPTION key until the boot manager screen appears. Select the Recovery HD and click on the downward pointing arrow button.

Reinstalling OS X Without Erasing the Drive

Repair the Hard Drive and Permissions: Upon startup select Disk Utility from the main menu. Repair the Hard Drive and Permissions as follows.

When the recovery menu appears select Disk Utility and press the Continue button. After Disk Utility loads select the Macintosh HD entry from the the left side list. Click on the First Aid tab, then click on the Repair Disk button. If Disk Utility reports any errors that have been fixed, then re-run Repair Disk until no errors are reported. If no errors are reported click on the Repair Permissions button. Wait until the operation completes, then quit Disk Utility and return to the main menu.

Reinstall OS X: Select Reinstall OS X and click on the Continue button.

Note: You will need an active Internet connection. I suggest using Ethernet if possible because it is three times faster than wireless.

OS X Mavericks- If your computer stops or delays while starting up with progress bar

That's just what I was going to say to BethHentges.

Ignore that Base System entry. Here's what you can try. Time Machine contains a hidden Recovery HD from which you can boot but you must use OPTION boot to get to it:

Boot Using OPTION key:

1. Restart the computer.

2. Immediately after the chime press and hold down the

"OPTION" key.

3. Release the key when the boot manager appears.

4. Select the disk icon for the Time Machine backup disk which may be named, Recovery HD.

5. Click on the arrow button below the icon.

This will boot into the Recovery Utilities Menu. Proceed as follows:

Erase the hard drive:

1. Select Disk Utility from the main menu and click on the Continue button.

2. After DU loads select your startup volume (usually Macintosh HD) from the

left side list. Click on the Erase tab in the DU main window.

3. Set the format type to Mac OS Extended (Journaled.) Click on

the Erase button and wait until the process has completed.

4. Quit DU and return to the main menu.

Reinstall OS X: Select Restore from Time Machine backup. Click on the Continue button.

Note: You will need an active Internet connection. I suggest using Ethernet if possible

because it is three times faster than wireless.

This should install the version of OS X that you had installed.

Yes. For help with doing individual file/folder restores from Time Machine please visit Pondini's Time Machine FAQ for help.