Number of primes less then $6000$ using $n/ \log n$ So I am trying to use this formula here and is giving me some trouble.

If I just substitute $6000$ into the formula, the answer is approximately $1500$.

But the number of primes under $6000$ is clearly half that. How do you use this formula properly?
$\pi(x)$ is the prime counting function, it gives the exact amount of primes below $x$

$$\pi(6000)=783$$

The formula provided is an approximation, so it gives:

$$\frac{6000}{\log_e 6000}=\frac{6000}{8.699514748}=689.69364083$$

While about $100$ from the actual answer, this is pretty accurate. It will increase in accuracy as $n$ goes to $\infty$

When you calculated the formula, you used $\log_{10}$ and not $\log_e$, that's why your value was so long from the actual value.
The prime number theorem doesn't say $\pi(n) = \frac{n}{\log n}$. It actually says $\lim_{n \to \infty} \pi(n) \large/ \frac{n}{\log n} = 1$, which means it gets more accurate as $n$ gets larger.