Question

If ${\log }_{k}x.{\log }_{5}k={\log }_{x}5,k\ne 1,k>0$, then $x$ is equal to

• A. $k$
• B. $\frac{1}{5}$
• C. $5$
• D. $-5$

Answer 1

Answer: (B), (C)

The correct options are
B $\frac{1}{5}$
C $5$

${\log }_{k}x.{\log }_{5}k={\log }_{x}5$

$\Rightarrow \frac{\log x}{\log k}\times \frac{\log k}{\log 5}=\frac{\log 5}{\log x}\cdots \cdots [\because {\log }_{b}a=\frac{\log a}{\log b}]\Rightarrow {\left(\log x\right)}^2={\left(\log 5\right)}^2\Rightarrow \log x=\pm \log 5$

$\Rightarrow x=5$ or $x=\frac{1}{5}$

Sm option $B$ and $C$ are correct.