IMac crashing randomly and slow boot time.

Question

My mid 2011 27" 2.7GHz i5 iMac has been crashing/powering off to black screen randomly. Apple just replaced the logic board and power supply last week. I have not had any more crashes yet, but on startup I get a white screen that will last for 1 whole minute at times before I finally get the Apple logo to appear. Can someone please tell me about how long my boot time should be and what could be the problem. Also my Mac keeps not seeing my External Hard drive Time Machine Backup. It will be working fine and then act like the drive is not connected. This seems to have started after Apple replaced my logic board and power supply. This is my first Mac ever and Im starting to get really frustrated!

Answer 1

When you see a beachball cursor or the slowness is especially bad, note the exact time: hour, minute, second.  
These instructions must be carried out as an administrator. If you have only one user account, you are the administrator.
Launch the Console application in any of the following ways:
☞ Enter the first few letters of its name into a Spotlight search. Select it in the results (it should be at the top.)
☞ In the Finder, select Go ▹ Utilities from the menu bar, or press the key combination shift-command-U. The application is in the folder that opens.
☞ Open LaunchPad and start typing the name.
The title of the Console window should be All Messages. If it isn't, select
          SYSTEM LOG QUERIES ▹ All Messages
from the log list on the left. If you don't see that list, select
          View ▹ Show Log List
from the menu bar at the top of the screen.
Each message in the log begins with the date and time when it was entered. Scroll back to the time you noted above.
Select the messages entered from then until the end of the episode, or until they start to repeat, whichever comes first.
Copy the messages to the Clipboard by pressing the key combination command-C. Paste into a reply to this message by pressing command-V.
The log contains a vast amount of information, almost all of it useless for solving any particular problem. When posting a log extract, be selective. A few dozen lines are almost always more than enough.
Please don't indiscriminately dump thousands of lines from the log into this discussion.
Please don't post screenshots of log messages—post the text.
Some private information, such as your name, may appear in the log. Anonymize before posting.