Hi I lost my admin passwords and i'm using MacBook Air which I can't use with a hard disc intall..

OS X 10.7 Lion, 10.8 MountainLion & 10.9 Mavericks

Reset Password using Recovery HD

Boot into Recovery Partition.

Start the computer,then press and hold down command and R keys to start into recovery partition.

When you see the Apple logo, release the keys.

Wait until OS X Utilities window shows up.

Move the mouse to the menubar at the top and click "Utilities", then select "Terminal"

from the drop down.

Terminal window will appear.

Type in resetpassword and press enter on the keyboard.

Leave the Terminal window open.

Reset Password Utility window will open with Macintosh HD selected.

Select the user account from the popup menu box.

Enter a new password.

Reenter the new password for the user.

Enter a hint.

Click the "Save" button.

Click  in the menubar and select Restart.

Log in.

When Keychain dialog box appears, select “Create New Keychain”.