Alternating Sum_{k>=1} (-1)^(k+1)*x^Log[k] = 0 polynomial in latex and root circle

Alternating Sum_{k>=1} (-1)^(k+1)*x^Log[k] = 0 polynomial in latex


For a sufficiently large $c$, what is known about $x$ in
$$1-x^{c\log (2)}+x^{c\log (3)}-x^{c\log (4)}+x^{c\log (5)}-x^{c\log (6)}+x^{c\log (7)}-x^{c\log (8)}+x^{c\log (9)}-\text{...}(-1)^{n+1}x^{c\log (n)}=0$$ as $n \rightarrow \infty$?
Do the roots in $x$ sit on a circle?
https://kconrad.math.uconn.edu/blurbs/galoistheory/numbersoncircle.pdf
https://math.stackexchange.com/q/4297761/8530
https://mathoverflow.net/q/364186/25104