Proof | Comment 1 of 2
Suppose log(3) is rational. Then, for some integers x and y, log(3)=x/y. Here is what we get.
ylog(3)=x
10^(ylog(3))=10^x
3^y=10^x
However, all powers of 3 are odd, and powers of 10 are even. Therefore, log(3) is irrational.

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