If $\text{log}_{\sqrt{3}} \space 27 = x$, then the value of x is

1. 3

2. 4

3. 6

4. 9

Given,

$\Rightarrow x = \text{log}${\sqrt{3}} \space 27 \\[1em] = \dfrac{\text{log}{10} \space 27}{\text{log}{10} \space \sqrt{3}} \\[1em] = \dfrac{\text{log}{10} \space 3^3}{\text{log}{10} \space 3^{\dfrac{1}{2}}} \\[1em] = \dfrac{3\text{log}{10} \space 3}{\dfrac{1}{2}\text{log}_{10} \space 3} \\[1em] = \dfrac{3}{\dfrac{1}{2}} \\[1em] = 3 \times 2 = 6.

Hence, Option 3 is the correct option.