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 * 89 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?
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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 http://crd-legacy.lbl.gov/~dhbailey/dhbpapers/digits.pdf, Bailey, Borwein, and Plouffe, On the rapid computation of various polylogarithmic constants.