I have recently come across a product of primes given by
$$ \prod_{p \geq 3}^{x}{\frac{p-2}{p-1}} = \prod_{p \geq 3}^{x}{\bigg(1-\frac{1}{p-1}\bigg)} $$
Taking the limit as $x \rightarrow \infty$, it can be easily seen that this product never reaches $0$, but also does not converge to any real value. Using some regression analysis on the product up to $x=100$, the product appears to be approximated by $\frac{1}{\ln x}$, specifically
$$ \prod_{p \geq 3}^{x}{\frac{p-2}{p-1}} \approx \frac{1}{B+A\ln x} $$ $$ A = 1 + \ln 2 $$ $$ B = \frac{1 + \ln 2}{2} $$
The actual regression calculator did not give such specific values for $A$ and $B$, though they were close enough for me to suspect a connection (especially considering the product skips the prime $2$). How can I verify that this growth rate is correct (and if it actually isn't, find the correct rate)? I have attempted replacing $p$ with the approximation of $n\ln n$, but to no avail.
The given product equals
$$ \exp\left[\sum_{p=3}^{x}\log\left(1-\frac{2}{p}\right)-\sum_{p=3}^x\log\left(1-\frac{1}{p}\right)\right]$$
or $$\exp\left[\sum_{p=3}^{x}\log\left(1-\frac{1}{p}\right)+\sum_{p=3}^x\log\left(1-\frac{1}{(p-1)^2}\right)\right] $$ where the second sum is convergent as $x\to +\infty$ and the behaviour of the first sum is known from Mertens' second theorem. The second sum is related to the twin prime constant $\Pi_2$. In particular the given product is approximately $ \exp\left[-\log\log x-B_1+\log \Pi_2\right]$ with $B_1$ being Mertens' constant.