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Use the unique factorization for integers theorem and the definition of logarithms to prove that $\log_3 (7)$ is irrational.

I am taking a beginners fundamental mathematics module, no advanced stuff please. Thanks!

My attempt.

Suppose for a contradiction that it is rational, that is $\log_3(7)=\frac{a}{b}$ for some $a,b\in R$ where $b\neq0$. Therefore, by the definition of logarithms, $7=3^{\frac{a}{b}}$. By the theorem of unique factorization, $7=1*7$ is unique.

Ok I'm plain stuck! Any help please?

Yellow Skies
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1 Answers1

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Use that $$\log_3(7)=\frac{a}{b} \iff b\log(7)=a\log(3) \iff \log(7^b)=\log(3^a) \iff 7^b=3^a ,$$ contradiction.

P..
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  • This is precisely the answer Jack provided in the comments. You should give Jack a few hours to post his comments as an answer. – JavaMan Oct 30 '12 at 18:27