Question

Prove that log3 to the base 2 , is not a rational number.

Answer 1

 .........Best Answer.........

Assume log(base 2)3 is rational 

This means that log(base 2)3 = a/b where a and b are integers 

2^(a/b) = 3 
2^a = 3^b 

Note that 2^a MUST be even if 'a' is an integer 
Note that 3^b MUST be odd if 'b' is an integer 

Hence a and b cannot both be integers 

Hence a/b is irrational 

Hence, using proof by contradiction, log(base 2)3 is irrational

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