We give algorithms for the computation of the d-th digit of certain
transcendental numbers in various bases. These algorithms can be easily
implemented (multiple precision arithmetic is not needed), require virtually
no memory, and feature run times that scale nearly linearly with the order of
the digit desired. They make it feasible to compute, for example, the billionth
binary digit of log (2) or Pi on a modest work station in a few hours run time.
We demonstrate this technique by computing the ten billionth hexadecimal
digit of Pi, the billionth hexadecimal digits of Pi^2; log(2) and log2^(2), and the
ten billionth decimal digit of log(9/10).
These calculations rest on the observation that very special types of identities
exist for certain numbers like Pi, Pi^2, log(2) and log2^(2). These are essentially
polylogarithmic ladders in an integer base. A number of these identities
that we derive in this work appear to be new.