Question

The value of $a={\log }_2\left({\log }_2\left({\log }_4\left(256\right)\right)\right)+2{\log }_{\sqrt{2}}\left(2\right)$ then $a$ is

• A. $1$
• B. $2$
• C. $3$
• D. $4$
• E. $5$

Answer 1

The correct option is E

$5$

Explanation for the correct option:

Find the value of $a$:

An equation $a={\log }_2\left({\log }_2\left({\log }_4\left(256\right)\right)\right)+2{\log }_{\sqrt{2}}\left(2\right)$ is given.

Since we know that ${\log }_a\left(a\right)=1$ and ${\log }_a{\left(c\right)}^b=b·{\log }_a\left(c\right)$.

Now, simplify the given equation as follows:

$$\begin{gathered} a={\log }_2\left({\log }_2\left({\log }_4{\left(4\right)}^4\right)\right)+2{\log }_{\sqrt{2}}{\left(\sqrt{2}\right)}^2 \\ \Rightarrow a={\log }_2\left({\log }_2\left(4{\log }_4\left(4\right)\right)\right)+4{\log }_{\sqrt{2}}\left(\sqrt{2}\right) \\ \Rightarrow a={\log }_2\left({\log }_2{\left(2\right)}^2\right)+4 \\ \Rightarrow a={\log }_2\left(2{\log }_2\left(2\right)\right)+4 \\ \Rightarrow a={\log }_2\left(2\right)+4 \\ \Rightarrow a=1+4 \\ \Rightarrow a=5 \end{gathered}$$

Therefore, the value of $a$ is $5$.

Hence, option (E) is the correct answer.