Can we simplify this logarithm? if so, please provides some tips

Question

${|x|^{11/10}} \log_{|x|^{{1/10}}}|x|$.

I only know doing the first step, not sure if it is correct

$\log_{|x|^{{1/10}}}(|x|^{|x|^{11/10}})$

as got stuck following this proof. Please help understand how we can get step two from step one.

Answer 1

If I interpret your question correctly, $V$ means the set of vertices and $E$ means the set of edges.

We have $|E|=O(|V|^2)$.

Hence, we can drop the first term.

Also, note that $\log_ab =\frac{\log b}{\log a}$.

Answer 2

Consider $$y=x^a\log_{x^b} (x)$$ Using the laws of logarithms $$y=\frac{x^a \log (x)}{\log \left(x^b\right)}=\frac{x^a \log (x)}{b\log \left(x\right)}=\frac 1b x^a$$