10.34 Find all values of $x$ for which the complex number $z=\left(\log _{0,5} \log _{4} \log _{8}\left(x^{2}+4\right)-1\right) i \quad$ is represented by a point lying in the upper part of the imaginary axis.
10.34 According to the condition, $\log _{0.5} \log _{4} \log _{8}\left(x^{2}+4\right)-1>0$ for any $x \in \boldsymbol{R}$. Therefore,

$\log _{0.5} \log _{4} \log _{8}\left(x^{2}+4\right)>\log _{0.5} 0.5 \Rightarrow$

$\Rightarrow 0<\log _{4} \log _{8}\left(x^{2}+4\right)<0.5 \Rightarrow$

$\Rightarrow 1<\log _{8}\left(x^{2}+4\right)<2 \Rightarrow$

$\Rightarrow 8<x^{2}+4<64 \Rightarrow$

$\Rightarrow 4<x^{2}<60$,

from which $2<x<2 \sqrt{15}$ or $-2 \sqrt{15}<x<-2$.

Answer: $-2 \sqrt{15}<x<-2$ or $2<x<2 \sqrt{15}$.