Q: MAC Air slow to boot after 10.9.2 update.

Try these in order testing your system after each to see if it's back to normal:

1. a. Resetting your Mac's PRAM and NVRAM

    b. Intel-based Macs: Resetting the System Management Controller (SMC)

2. Restart the computer in Safe Mode, then restart again, normally. If this doesn't help, then:

     Boot to the Recovery HD: Restart the computer and after the chime press and hold down the

     COMMAND and R keys until the Utilities 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.

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

When the recovery menu appears select Disk Utility. After DU loads select your hard drive entry (mfgr.'s ID and drive size) from the the left side list.  In the DU status area you will see an entry for the S.M.A.R.T. status of the hard drive.  If it does not say "Verified" then the hard drive is failing or failed. (SMART status is not reported on external Firewire or USB drives.) If the drive is "Verified" then select your OS X volume from the list on the left (sub-entry below the drive entry,) click on the First Aid tab, then click on the Repair Disk button. If DU 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 DU and return to the main menu. Select Restart from the Apple menu.

     Reinstall the 10.9.2 update: OS X Mavericks 10.9.2 Update (Combo).

4. Reinstall Lion/Mountain Lion, Mavericks: Reboot from the Recovery HD. Select Reinstall Lion/Mountain Lion, Mavericks from the Utilities menu, 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.

Reinstall the 10.9.2 update: OS X Mavericks 10.9.2 Update (Combo).

I've got the similiar problem. Except for me it's not as extreme, the boot time takes 1-2min, while normally (before update), as OP mentioned, it took around 10s.

What's interesting, I have 2011 MBA as well.