Question

If ${\log }_5{\log }_5{\log }_{2}x=0$, then the value of $x$ is

• A. $32$
• B. $125$
• C. $625$
• D. $25$

Answer 1

The correct option is A

$32$

Explanation for the correct answer:

The given expression can be written as ${\log }_5\left({\log }_5\left({\log }_{2}x\right)\right)=0$

Using the property of logarithms

${\log }_a\left(1\right)=0$ for any real value of $a$ we can write as

${\log }_5\left({\log }_5\left({\log }_{2}x\right)\right)={\log }_{5}1$

$\Rightarrow$ ${\log }_5\left({\log }_{2}x\right)=1$ $\left[\because {\log }_{m}a={\log }_{m}b\Rightarrow a=b\right]$

$\Rightarrow$ ${\log }_{2}x=5^1=5$ $\left[\because {\log }_{a}b=m\Rightarrow b=a^m\right]$

$\Rightarrow$ ${\log }_{2}x=5\times 1$

$\Rightarrow$ ${\log }_{2}x=5\times {\log }_{2}2$ $\left[\because {\log }_{a}a=1\right]$

$\Rightarrow$ ${\log }_{2}x={\log }_2{2}^5$ $\left[\because {\mathrm{mlog}}_{a}b={\log }_a{b}^m\right]$

$\Rightarrow$ ${\log }_{2}x={\log }_{2}32$

$\Rightarrow$ $x=32$ $\left[\because {\log }_{m}a={\log }_{m}b\Rightarrow a=b\right]$

Hence, the value of $x$ is $32$.

Hence, option (A) i.e. $32$ is the correct answer.