Question

The solution of $5^{{\log }_{a}x}+5x^{{\log }_{a}5}=3(a>0,a\ne 1)$ is

• A. $2^{{\log }_{a}5}$
• B. $2^{-{\log }_{a}5}$
• C. $2^{{\log }_{5}a}$
• D. $2^{-{\log }_{5}a}$

Answer 1

The correct option is D $2^{-{\log }_{5}a}$

We have,
$5^{{\log }_{a}x}+5x^{{\log }_{a}5}=3$
$\Rightarrow x^{{\log }_{a}5}+5x^{{\log }_{a}5}=3[\because x^{{\log }_{a}y}=y^{{\log }_{a}x}]$
$\Rightarrow 6\cdot x^{{\log }_{a}5}=3$
$\Rightarrow x^{{\log }_{a}5}=\frac{1}{2}\Rightarrow x=(2^{-1}{)}^{{\log }_{5}a}=2^{-{\log }_{5}a}$