logical deduction - How long has the warden been the warden?

Summary:

It is your first day in prison serving a literal life-sentence under the supervision of the warden. You are to be kept in solitary confinement with no additional information for the duration of your stay.

However, there is one chance of escape. The warden will set you free if you guess the exact number of days (including the day you guess) it has been since he first became warden. The catch is that if you guess the wrong number of days, you will be executed immediately.

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Rules:

1. You may guess the exact number of days the warden has been in his position at any time and accept the consequences (this does not count as a yes/no question).


2. You may ask n+1 total yes/no questions as long as it has been 2^n days since the warden's first day. For example:
   
   
   
   
   1. If it is his first day on the job, you get 1 Y/N question
   
   
   2. If it is his 2nd-3rd day on the job, you get 2 Y/N question
   
   
   3. If it is his 4th-7th day on the job, you get 3 Y/N questions
   
   
   4. If it is his 8th-15th day on the job, you get 4 Y/N questions
   
   
   5. etc.
   
   
   
   


3. If you ask the warden too many yes/no questions, you will be executed. For example:
   
   
   
   
   1. If you ask your 2nd Y/N question and he had his job less than 2 days, you are executed
   
   
   2. If you ask your 3rd Y/N question and he had his job less than 4 days, you are executed
   
   
   3. If you ask your 4th Y/N question and he had his job less than 8 days, you are executed
   
   
   4. If you ask your 5th Y/N question and he had his job less than 16 days, you are executed
   
   
   5. etc.
   
   
   
   

That's it.

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Question:

What strategy would you use and why?