I noticed that there are many pairs of primes $p_1, p_2$ such that $$\frac {\log(p_1/p_2)}{ \log(2)} \approx \frac{N}{69324}$$ for some natural N with an error of order $10^{-8}$ . I found around 20 pairs, but more can be found if one has the patience.
Other such sets with other common denominators can likewise be found, as can sets with $\log(3)$ or $\log(5)$ in place of $\log(2)$ in the denominator slot of the LHS.
I would like to know whether this is some interesting property of primes, or whether it is statistically unremarkable and expected through pure chance. It would certainly be unremarkable if each ratio had its own denominator, but here they all have $69324.$
$$ \begin{array}{ccccc} \text{Prime $\#$1} & \text{Prime $\#$2} & \text{Numerator, N } & \text{Denominator, D} & \text{Error= }\frac{\log \left(\frac{\text{p1}}{\text{p2}}\right)}{\log (2)}-\frac{N}{D} \\ 461 & 3 & 503546 & 69324 & \text{-5.305e-8} \\ 617 & 3 & 532697 & 69324 & \text{-3.599e-8} \\ 1201 & 3 & 599310 & 69324 & \text{-5.359e-8} \\ 661 & 5 & 488497 & 69324 & \text{-1.893e-8} \\ 1567 & 5 & 574825 & 69324 & \text{-3.483e-8} \\ 449 & 7 & 416167 & 69324 & \text{-6.65e-8} \\ 1367 & 7 & 527517 & 69324 & \text{-1.173e-8} \\ 1453 & 7 & 533619 & 69324 & \text{-1.613e-8} \\ 149 & 11 & 260640 & 69324 & \text{1.417e-8} \\ 1901 & 11 & 515293 & 69324 & \text{9.967e-9} \\ 401 & 23 & 285885 & 69324 & \text{-1.567e-8} \\ 191 & 29 & 188523 & 69324 & \text{2.263e-8} \\ 479 & 31 & 273808 & 69324 & \text{7.581e-10} \\ 461 & 41 & 242015 & 69324 & \text{2.242e-8} \\ 1559 & 61 & 324136 & 69324 & \text{-3.233e-9} \\ 881 & 67 & 257671 & 69324 & \text{1.148e-8} \\ 449 & 71 & 184459 & 69324 & \text{-1.884e-8} \\ 887 & 73 & 249772 & 69324 & \text{-4.843e-8} \\ 317 & 79 & 138964 & 69324 & \text{-2.396e-8} \\ 1019 & 107 & 225405 & 69324 & \text{-2.148e-9} \\ 641 & 193 & 120050 & 69324 & \text{8.11e-9} \\ 787 & 197 & 138521 & 69324 & \text{-1.657e-8} \\ \end{array} $$