Question

If ${\log }_5\left(a\right).{\log }_a\left(x\right)=2$, then $x$ is equal to

• A. $5$
• B. $125$
• C. $25$
• D. None of these

Answer 1

The correct option is C

$25$

Explanation for the correct option:

Given: ${\log }_5\left(a\right).{\log }_a\left(x\right)=2$

$$\begin{gathered} \Rightarrow \frac{\log \left(a\right)}{\log \left(5\right)}·\frac{\log \left(x\right)}{\log \left(a\right)}=2\left[\because {\log }_a\left(b\right)=\frac{\log \left(b\right)}{\log \left(a\right)}\right] \\ \Rightarrow \frac{\log \left(x\right)}{\log \left(5\right)}=2 \\ \Rightarrow {\log }_5\left(x\right)=2\left[\because {\log }_a\left(b\right)=\frac{\log \left(b\right)}{\log \left(a\right)}\right] \\ \Rightarrow x=5^2\left[\because {\log }_a\left(b\right)=c\Rightarrow b=a^c\right] \\ \Rightarrow x=25 \end{gathered}$$

Hence option(C) i.e. $25$ is correct.