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?
Best Answer
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.