Does $\log\sqrt x$ grow as fast as $4\log x$ or slower? Based on their function diagram, I would guess that $\log\sqrt x$ is slower. However $$\lim_{ x\to \infty} \frac {\log (\sqrt x)}{\log x} = 1/8$$
So we can conclude that $\log\sqrt x$ grows as fast as $4\log x$. How can I solve this problem?
$$\log \sqrt{x} = \log x^{1/2} = \frac{1}{2} \log x.$$ So the ratio $$\frac{4 \log x}{\log \sqrt{x}} = \frac{4}{1/2} = 8.$$ So they are both $O(\log x)$ but obviously one is a constant multiple of the other.