Is the power of a natural number logspace-computable? Given a constant $n\in\mathbb{N}_+$, is the simple power function
$$
λ\,x\in\mathbb{N}_+.\,x^n
$$
logspace computable by a logspace transducer (which has a read-only input tape, a working read-write tape (on which the transducer is allowed to use $\mathcal{O}(\log (\text{input size}))$) cells, and a write-only output tape), assuming that the input and the output are in binary?  (It is not even clear to me how to do it for $n=2$ in logspace.) Any ideas or references?
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.