$\log_{\sqrt{2}} \space \Big(4\sqrt{2}\Big) =$

1. 8

2. 6

3. 4

4. 5

Given,

$\Rightarrow \log_{\sqrt{2}} \space \Big(4\sqrt{2}\Big)$

Let,

$\Rightarrow \log_{\sqrt{2}} \Big(4\sqrt{2}\Big) = y \\[1em] \Rightarrow (\sqrt{2})^{y} = 4\sqrt{2} \\[1em] \Rightarrow [(2)^{\dfrac{1}{2}}]^{y} = 2^2 \times 2^{\dfrac{1}{2}} \\[1em] \Rightarrow (2)^{\dfrac{y}{2}} = 2^{2 + \dfrac{1}{2}} \\[1em] \Rightarrow (2)^{\dfrac{y}{2}} = 2^{\dfrac{4 + 1}{2}} \\[1em] \Rightarrow (2)^{\dfrac{y}{2}} = 2^{\dfrac{5}{2}} \\[1em] \Rightarrow \dfrac{y}{2} = \dfrac{5}{2} \\[1em] \Rightarrow y = \dfrac{5}{2} \times 2 \\[1em] \Rightarrow y = 5.$

Hence, option 4 is the correct option.