Proving question for Logarithm

Question

$\log_2 x + \log_x 2 + 2$ in the form of $\frac{(a+b)^2}{a}$

Answer 1

Note that this is equal to $$\log_2x+\frac{1}{\log_2x}+2=(\sqrt{\log_2x})^2+\left(\sqrt{\frac{1}{\log_2x}}\right)^2+2=$$ Now use $(\sqrt{a})^2+\left(\frac{1}{\sqrt{a}}\right)^2+2=\left(\sqrt{a}+\frac{1}{\sqrt{a}}\right)^2=\frac{\left(a+1\right)^2}{a}$

Answer 2

$\log_2(x)+\log_x(2)+2 = \log_2(x)+\frac{\log_2(2)}{\log_2(x)}+2 = \log_2(x)+\frac{1}{\log_2(x)}+2$

Setting $a=\sqrt{\log_2(x)}$ this is $(a+\frac{1}{a})^2$