How does $\left(\log \sqrt x\right)^2 = \frac 14(\log x)^2\;?$

Question

So as the title says it all:

How does $\;\left(\log \sqrt x\right)^2 = \frac 14(\log x)^2 \;?$

To be specific, why the removal of root, and how do we get 4 in denominator?

Answer 1

Recall: $$\log\left(a^b\right) = b\log a$$

Here, that means that $$\log (\sqrt x) = \log x^{1/2} = \frac 12 \log x$$

So $$\left(\log \sqrt x\right)^2 = \underbrace{\left(\frac {\log x}2\right)^2= \frac {(\log x)^2}{2^2}}_{\large \left(\frac ab\right)^c = \frac{a^c}{b^c}} =\frac 14(\log x)^2$$