At page 65 of Rosemberger's book there is a proof that for any natural number x>=21:
$\pi(x)>\frac {lnx}{2lnlnx}$ where ln is the natural log. The proof goes as follows: For fixed x let $p_i$ run all over the primes less than or equal to x. Then from the FTA (fundamental theorem of arithmetic) the number of integral solutions to the inequality: $$\prod_{\substack{p_i}}{p_i^{e_i}}<=x$$ for $e_i>=0$ is precisely x. Ok until now I have understood. Now I don't understand well this passage: on the other hand the number of solutions is the product of the number of choices for each $e_i$. What does it mean? Then since for a solution $p_i^{e_i}$<=x we have: $$e_i<=1+\frac{lnx}{lnp_i}<=1+\frac{lnx}{ln2}<(lnx)^{2}$$...now I don't understand why for x>20 we have that: $$x<=\prod_{\substack{p_i}}{(1+\frac{lnx}{lnp_i})}$$?