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?

• How would it work in log space if the size of the output is exponential in the size of the input? I know you specified a transducer but still the computation will take exponential size to reach it. I am not saying it does not work I have no experience in that area but I find it very interesting if it works – narek Bojikian Oct 26 '19 at 19:02
• @narekBojikian The size of the input is roughly $\log x$, whereas the size of the output is roughly $\log(x^n) = n\log x$, i.e., linear in the size of the input. – user111225 Oct 26 '19 at 20:15
• Assuming n is a constant? – narek Bojikian Oct 26 '19 at 20:16
• @narekBojikian Yes. The function to be computed has only $x$ as input, not $n$. I've just made an edit to emphasize this. – user111225 Oct 26 '19 at 20:17
• But then the trivial method is already log space isnt it? I mean we never grow above the output and you said the output is small enouph – narek Bojikian Oct 26 '19 at 20:19