I have been trying to prove a certain claim and have hit a wall. Here is the claim...
Claim: If $n$ is a positive integer then $\log_{7}n$ is an integer or it is irrational
Proof (so far): Let $y=\log_{7}n$. Note that to say $n$ is a positive integer is equivalent to saying that n is a non-zero natural number. We will proceed by trying to prove the contrapositive.
Claim (alternate version): If $y$ is a rational number and is not an integer, then either $n$ is zero or it is not a natural number.
Given the above we can assume that there exist integers $a$ and $b$ such that $y$ equals the quotient of $a$ over $b$. We can also assume from the premises that $b$ does not equal one and that $a$ and $b$ are relatively prime. Note thus that $n$ may be considered equal to seven raised to the power of $a$ over $b$. Further note that because of this $n$ cannot be zero or negative. To prove the claim, one must prove that $n$ is not a natural number.
Where I am stuck: How can I guarantee from here that $n$ is not a natural number? Is there any way to concretely ensure that there are no integers $a$ and $b$ such that the fractional exponent above will never give an integer when raising seven to its power?
I have been trying to play around with a proof that there is no such thing as a rational root of a prime number, but that hasn't shook anything loose so far.
\log_7n
meant to be $\log_7 n$? $\endgroup$