$\lfloor\log{p_{n}}\rfloor$ having more than one solution for individual $k$

Question

Question: If you assume that

a. $k\in\Bbb{N}$

b. $p_n$ denotes the $n$'th prime number. $p_0$ doesn't exist.

c. $n\in\Bbb{N}$

I am fairly certain that:

At least two distinct integer values for $n$ satisfy $\lfloor\log{p_n}\rfloor=k$ for any $k\geqslant 2$

I am looking for a proof that conjecture.

Attempts: I have done two things and then I hit my roadblock. (It is amazingly hard to elaborate on them because I am on my iPhone here)

a) I have managed to prove that there are at least one solution for each $k$ by comparing the behavior of $\log$ and $\sqrt[3]{/space}. I used this [paper][1].

b) I also used the brute force method. Going up to $k=777$, there are at least two $n$ solutions.

Thanks for any help.

[1][http://arxiv.org/pdf/0810.2113v2.pdf]

Answer 1

Hint: Use strengthenings of Bertrand's Postulate, in particular Nagura's result that for $n\ge 25$ there is always a prime between $n$ and $n\left(1+\frac{1}{5}\right)$. We can use this to show that unless $k$ is very small, there are always at least $4$ distinct primes $p$ that satisfy $\lfloor \log p \rfloor=k$.