Let $\displaystyle p,q$ be primes. Define the function $\displaystyle \sigma:\mathbb{N} \to \mathbb{N}$ by $\displaystyle \sigma(n) = k$ where $\displaystyle k = \min \{a \in \mathbb{N} \mid a> n \log_p{q} \}$. Define the function $\displaystyle \tau:\mathbb{N} \to \mathbb{N}$ by $\displaystyle \tau(n) = p^{\sigma(n)} - q^n$. Does anyone have a suggestion for how I might approximate $\displaystyle \tau(n)$?