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.

$$\log_23+\log_34>2\sqrt2$$ because it's
$$\log_23+\frac{2}{\log_23}>2\sqrt2,$$ which is AM-GM: $\log_23+\frac{2}{\log_23}>2\sqrt{\log_23\cdot\frac{2}{\log_23}}=2\sqrt2$  or
$$\log^2_23-2\sqrt2\log_23+2>0$$ 0r
$$\left(\log_23-\sqrt2\right)^2>0,$$
which is obvious.
