Hamming weight of powers
This answer expands my comment above.

You can do it with $O(\log e + \log \log b)$ space as follows:

1) First compute $b^e$ in Chinese remainder representation modulo sufficiently many primes.

2) Then use the Chiu-Davida-Litow algorithm to convert the Chinese remainder representation into binary representation. (Informatique Theoretique et Applications, Vol 35(3), pages 259-275, 2001)

3) Finally, just count the number of $1$'s.

This is a composition of a finite number of log-space computable functions, which is itself log-space computable.