Question

Show that log 7 n is either an integer or irrational, where n is a positive integer. use whatever familiar facts about integers and primes you need, but explicitly state such facts

Answer 1

Suppose log7n=pq is rational,

Then 7p/q=n,

Raising both sides to the qth power,

We see that 7p=nq.

Now we have by unique prime factorization that n=7k for some integer k,

Since it divides 7p.

But then 7p=7kq, or p=kq, but then pq=k is an integer as desired.