Vinogradov's "Elements of Number Theory", Ch1, Ex6: what is he talking about?

Assume that there are only finitely many primes $p_1,\ldots, p_k$ Then there are at most $$\left(\frac{\log(N)}{\log(p_1)}+1\right) \cdot ... \cdot \left(\frac{\log(N)}{\log(p_k)}+1\right) \le \left(2\frac{\log(N)}{\log(p_1)}\right) \cdot ... \cdot \left(2\frac{\log(N)}{\log(p_k)}\right) \tag{1}$$ numbers whose canonical decomposition includes only $\{p_1,\ldots, p_k\}$ that don't exceed $N$, if $N \ge \max\{p_1,\ldots,p_k\}$.

Therefore there are at most $$2^kt^k\frac{\log(N)}{\log(p_1)} \cdot ... \cdot \frac{\log(N)}{\log(p_k)} \tag{2}$$ numbers whose canonical decomposition includes only $\{p_1,\ldots, p_k\}$ that don't exceed $N^t$. But

$$\lim_{t \to \infty}2^kt^k\frac{\log(N)}{\log(p_1)} \cdot ... \cdot \frac{\log(N)}{\log(p_k)}/ N^t=0$$