Message expansion in NTRU: why is it $\log_p(q)$-to-$1$?
I guess you are taking this information from this document. 

In Section 2.1 you can see a table with different sizes. In particular, a plaintext block (that is, an encoded message) has size $(n-k) \log_2 p$ bits, while a ciphertext has size $n \log_2 q$ bits. The explanation is simple: ciphertexts are actually polynomials of $n$ terms (since degree is $n-1$), where each coefficient can be represented with $\log_2 q$ bits, while encoded messages are polynomials of $n-k$ terms (since degree is $n-k-1$),  where each coefficient can be represented with $\log_2 p$ bits.

Now, the "message expansion" (also known as "ciphertext expansion") is simply the ratio between the ciphertext size and the plaintext size. In other words, it is an indication of how much a ciphertext increases with respect to the original message. Ideally, you don't want it to expand too much for obvious reasons, although it can be shown that some expansion must exist for the encryption to be semantically secure (in the case of public-key schemes, like NTRU). In this case, it is $\frac{n \log_2 q}{(n-k) \log_2 p}$. Finally, by the properties of logarithms we have that $\frac{\log_2 q}{\log_2 p} = \log_p q$, so the final message expansion is $\frac{n}{n-k}\log_p q$.