SOLUTION: Prove by contradiction log2 11 is irrational. I'm a little stuck. Thank you.

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Question 1206127: Prove by contradiction log2 11 is irrational. I'm a little stuck. Thank you.
Answer by ikleyn(52787) About Me  (Show Source):
You can put this solution on YOUR website!
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Prove by contradiction log2 11 is irrational. I'm a little stuck. Thank you.
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Let assume that  log%282%2C%2811%29%29  is a rational number

    log%282%2C%2811%29%29 = m%2Fn,


where m and n are integer numbers.  Then

    2%5E%28m%2Fn%29 = 11.


Raise both sides of this equation to degree n.  You will get

    2%5Em = 11%5En.    (*)


But 2 and 11 are relatively prime numbers.

So, equality (*) CONTRADICTS to the basic theorem of arithmetic
about uniqueness of the decomposition of integer numbers into a product of primes.

The proof is complete.