Question

If ${\log }_k\left(x\right)·{\log }_5\left(k\right)={\log }_x\left(5\right),k\ne 1,k>0$, then $x$ is equal to:

• A. $k$
• B. $0$
• C. $5$
• D. None of these

Answer 1

The correct option is C

$5$

Explanation for the correct option:

Given ${\log }_k\left(x\right)·{\log }_5\left(k\right)={\log }_x\left(5\right)$ where$k\ne 1,k>0$

We know that, ${\mathbf{log}}_{\mathbf{b}}\left(a\right)\mathbf{=}\frac{\mathbf{log}\mathbf{\left(}\mathbf{a}\mathbf{\right)}}{\mathbf{log}\mathbf{\left(}\mathbf{b}\mathbf{\right)}}$

So,

$$\begin{gathered} \frac{\log \left(x\right)}{\cancel{\log \left(k\right)}}·\frac{\cancel{\log \left(k\right)}}{\log \left(5\right)}=\frac{\log \left(5\right)}{\log \left(x\right)} \\ \Rightarrow {\left(\log \left(x\right)\right)}^2={\left(\log \left(5\right)\right)}^2 \\ \Rightarrow \log \left(x\right)=\log \left(5\right) \\ \Rightarrow x=5 \end{gathered}$$

Hence, option(C) i.e.$5$, is the correct answer.