Question related to \left\{p\leq x| \ \text{the proprities $A_1(p), A_2(p),\ldots$ hold}\right\}.

Question

It is not difficult to see that $$\#\left\{p\leq x| \ \text{$p$ is prime and the proprities $A_1(p), A_2(p),\ldots, A_k(p)$ hold}\right\}<\frac{x}{\log x}.$$ Ma questions are: $$1) \#\left\{p\leq x| \ \text{$p$ is prime and the proprities $A_1(p), A_2(p),\ldots, A_k(p)$ hold}\right\}\sim O(\frac{x}{\log^k x})?$$ 2) Can we use sieve theory to prove that? Thank you