According to Mertens Theorem $$ \lim_{n \to \infty } (\frac{1}{ln(p_n)} \prod_{k=1}^n \frac{1}{1-\frac{1}{p_k}})=1.781072\dots$$
so we can say $$ 1.78 \ ln(p_n) \sim \prod_{k=1}^n \frac{1}{1-\frac{1}{p_k}}$$
As $p_n \sim n \ ln(n)$ I tried to plot both side in Desmos 
but it seems that they are not going to be similar I tried other number like 6.6 and it was better but why such things happened ?