Prove that the number x that satisfy 2 ^ x = 5 is not a number rasional.
please help me
please help me
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Ususally you solve a problem like this by assuming x is the rational a/b, where a and b are integers, and trying to find a contradiction.
2 ^(a/b) = 5
=> (2^a) ^ (1/b) = 5
=> 2^a = 5^b
See the contradiction? (Remember a and b are integers).
EDIT: p.s. one extra detail: x is positive, so if it was rational there would exist positive a and b. (The contradiction isn't so obvious if negative powers are allowed).
2 ^(a/b) = 5
=> (2^a) ^ (1/b) = 5
=> 2^a = 5^b
See the contradiction? (Remember a and b are integers).
EDIT: p.s. one extra detail: x is positive, so if it was rational there would exist positive a and b. (The contradiction isn't so obvious if negative powers are allowed).