Question

If$A={\log }_2{\log }_2{\log }_{4}256+2{\log }_{\sqrt{2}}2$, then$A$ is equal to

• A. $2$
• B. $3$
• C. $5$
• D. $7$

Answer 1

The correct option is C

$5$

Explanation for correct option:

Finding the value by using properties of the logarithm:

${\log }_m{n}^p=p{\log }_{m}n$ and ${\log }_{a}a=1$

$$\begin{gathered} A={\log }_2{\log }_2{\log }_{4}256+2{\log }_{\sqrt{2}}2 \\ \Rightarrow {\log }_2{\log }_2{\log }_4{4}^4+2{\log }_{\sqrt{2}}{\left(\sqrt{2}\right)}^2 \\ \Rightarrow {\log }_2{\log }_{2}4{\log }_{4}4+4{\log }_{\sqrt{2}}\sqrt{2}\mathbf{\because }\mathit{l}\mathit{o}{\mathit{g}}_{\mathbf{m}}{\mathit{n}}^{\mathbf{p}}\mathbf{=}\mathit{p}\mathit{l}\mathit{o}{\mathit{g}}_{\mathbf{m}}\mathit{n} \\ \Rightarrow {\log }_2{\log }_{2}4\times 1+4\times 1\mathbf{\because }\mathbf{}\mathit{l}\mathit{o}{\mathit{g}}_{\mathbf{a}}\mathit{a}\mathbf{}\mathbf{=}\mathbf{1} \\ \Rightarrow {\log }_2{\log }_2{2}^2+4 \\ \Rightarrow {\log }_{2}2{\log }_{2}2+4 \\ \Rightarrow 1\times 1+4=5 \end{gathered}$$

Hence,$A=5$ Option $\mathbf{\left(}\mathit{C}\mathbf{\right)}$ is correct .