Question

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

• A. $k$
• B. $\frac{1}{5}$
• C. $5$
• D. none of these

Answer 1

Answer: (B), (C)

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

$\text{Given,}{\log }_{k}x{\log }_{5}k={\log }_{x}5\Rightarrow \frac{\log x\log k}{\log k\log 5}=\frac{\log 5}{\log x}\Rightarrow {\left(\log x\right)}^2={\left(\log 5\right)}^2\Rightarrow \log x=\pm \log 5=\log 5,\log 5^{-1},[n\log a=\log a^n]\therefore x=5,\frac{1}{5}$.