Is there a binary spigot algorithm for log(23) or log(89)? The Bailey-Borwein-Plouffe formula yields a binary spigot algorithm for π, and related formulas give the bits of log(2) and those of the logarithms of some other integers. I got stuck (over a year ago, so none of this is fresh in my head) trying to find a spigot algorithm for the bits of log(23), and I guess that the difficulty is because $23 \times 89 = 2^{11}-1$ is a Mersenne number. Is any binary spigot algorithm known for log(23) or log(89) which is just as fast as those for π and log(2)? If not, is there any reason to think that one doesn't exist?

Dang-Khoa Do, Spigot algorithm and reliable computation of natural logarithm, Reliable Computing 10 (2004) 489-500, gives spigot algorithms for computing various logarithms. I didn't look at it closely enough to tell whether $\log23$ is amenable to Do's methods.

EDIT: Incidentally, OP is quite correct to relate the difficulty to the Mersenne connection. See the top of page 11 of < Bailey, Borwein, and Plouffe, On the rapid computation of various polylogarithmic constants.