  BBP-type formulas are usually discovered experimentally, one at a time and in
specific bases, through computer searches. In this paper, however, we derive
directly, without doing any searches, explicit digit extraction BBP-type
formulas in general binary bases $b=2^{12p}$, for $p$ positive odd integers. As
particular examples, new binary formulas are presented for $\pi\sqrt 3$,
$\pi\sqrt 3\log 2$, $\sqrt 3\;{\rm Cl}_2(\pi/3)$ and a couple of other
polylogarithm constants. A variant of the formula for $\pi\sqrt 3\log 2$
derived in this paper has been known for over ten years but was hitherto
unproved. Binary BBP-type formulas for the logarithms of an infinite set of
primes and binary BBP-type representations for the arctangents of an infinite
set of rational numbers are also presented. Finally, new binary BBP-type zero
relations are established.
