Question

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

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

Answer 1

Answer: (B), (C)

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

${\log }_{k}x\cdot {\log }_{5}k={\log }_{x}5;k>0,k\ne 1$
$\Rightarrow \frac{\log x\cdot \log k}{\log k\cdot \log 5}={\log }_{x}5$
$\Rightarrow \frac{\log x}{\log 5}={\log }_{x}5$
$\Rightarrow {\log }_{5}x=\frac{1}{{\log }_{5}x}$
$\Rightarrow ({\log }_{5}x{)}^2=1$
$\Rightarrow {\log }_{5}x=\pm 1$
$\Rightarrow x=\frac{1}{5},5$