Proof verifiation: $\log_a b$ is either an integer or irrational number

Question

My idea is to show that $\log_b a$ cannot be written as a fraction if it is not an integer. Here is what I tried:

Write $\log_b a$ as follows: $$\log_b a = \frac{p}{q}$$ Where $p,q$ are in their lowest form. By definition: $$b^{\frac{p}{q}} = a$$ Raising to the $b$^th power: $$b^p = a^q$$ Now, write $b,a$ in their unique prime factorization raising them to $p$ and $q$ then cancel common primes out. If the result is $1=1$ then we are done. Otherwise, by Euclid Lemma These two numbers cannot be equal. Hence, there are no integers $p,q$ such that $b^p = a^q$.

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I want to check my method above.

Answer 1

What you are trying to prove is false.

For example,

$$\log_4 8 = \frac{3}{2} $$

Answer 2

On the country there are many such logarithms which are rationals. For example $$ log _{4} 2 =1/2$$ In general $$\log_{2^k} 2=1/k$$