Prove that log 36 ⁡ 30 is irrational number.

Question

Prove that       log          36        ⁡  30 is irrational number.

Marlee Norman · Accepted Answer

In the equality      36    p    =      30    q   as 1)   36 as it ends in   6 ,   36 to the power anything always ends in   6 ,2)   30 as it ends in   0 ,   30 to the power anything always ends in   0 ,so differing units digit itself proves that for no integer this equality is possible except for p and q as   0 , but   q  ≠  0 as per definition of rationality , hence proved that p,q doesn&#039;t exist for this equality

oleifere45 · Accepted Answer

So, we have   (      6    2        )    p    =  (  6  ⋅  5      )    q      ⟹        6          2      p      −      q        =      5    q  As   (  6  ,  5  )  =  1 we need   q  =  2  p  −  q  =  0 which is impossible as   q  ≠  0

Prove that log

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