Is the power of a natural number logspace-computable?
Yes, iterated multiplication is even in uniform TC0:

Hesse, William; Allender, Eric; Mix Barrington, David (2002). "Uniform constant-depth threshold circuits for division and iterated multiplication". Journal of Computer and System Sciences. 65: 695–716. doi:10.1016/S0022-0000(02)00025-9.

Explanation: We have TC0 ⊆ NC1 ⊆ L, where TC0 denotes the class of Boolean circuits with constant depth and polynomial size, where unbounded fan-in threshold gates are allowed, NC1 denotes the class of Boolean circuits with logarithmic depth and polynomial size, and L denotes logspace.