We know that the number of primes in a given interval $(0,x)$ is about $x/logx$. Now, I have an inequality of the form $p_{n}≤x$ where $p_{n}$ is the $n$-th prime. Then I deduce that $n<x/logx$
My question is: Is this estimation correct and how I can measure the corresponding error.
In general this is not right. For example, if n = 4: $ p_4 = 7 < 8 $, however 4 > 8 / log(8).
In your terms (n is number of primes up to x), this what was proved:
$$ \lim_{x\to \infty } \frac{n * log(x)}{x} = 1 $$
So it's not a good idea to compare n and x / log(x). The only you can say, for any small d you can find large numbers x that n will be less than (1 + d)(x / log(x))
