Which of the following is bigger (logarithms) I need to compare those two expressions and decide which is bigger.
$2 \sqrt2$ or $\log_2(3)+\log_3(4) $.
So I tried to simplify so the log expression so I know
and so
$$ \log_2(4) \times (\log_4(3) + \log_3(2)) ?? 2 \times \sqrt2$$ 
and then 
$$2 \times \log_2(2)\times(\log_4(3)+\log_3(2)) ?? 2 \sqrt2$$
$$\log_2(2) \times (\log_4(3)+\log_3(2)) ?? \sqrt2 $$
and I know $\ log_2(2) = 1$ so now I need to compare those two expressions:
$$\log_4(3)+\log_3(2) $$against$$  \sqrt2 $$
I'm not really sure what i'm doing wrong here.

$\log_3 4 = \dfrac{\log_2 4}{\log_2 3} = \dfrac{2}{\log_2 3}$
$A := {\log_2 3}+ \log_3 4  = {\log_2 3} + \dfrac{2}{\log_2 3}$
Dividing by$A$ by  $\sqrt 2$, observe $ \dfrac{\log_2 3}{\sqrt 2} + \dfrac{\sqrt 2}{\log_2 3} > 2$ by AM-GM inequality (since ${\log_2 3 \ne \sqrt 2}$)
Thus $A>2\sqrt 2$
