Question

The value of $x$ that satisfies the expression $5^{{\log }_{a}x}+5x^{{\log }_{a}5}=3(a>0)$ is

• A. $x=2^{-{\log }_{5}a}$
• B. $x=2^{{\log }_{5}a}$
• C. $x=4^{-{\log }_{25}a}$
• D. $x=4^{{\log }_{25}a}$

Answer 1

The correct option is A $x=2^{-{\log }_{5}a}$

Given, $5^{\log x}+5x^{\log 5}=3$
$\Rightarrow 5^{{\log }_{a}x}+{5.5}^{{\log }_{a}x}[\because a^{{\log }_{b}c}=c^{{\log }_{b}a}]=3\Rightarrow 5^{{\log }_{a}x}=\frac{1}{2}$
Taking $\log$ both side on base a.
$\Rightarrow {\log }_{a}x{\log }_{a}5={\log }_a{2}^{-1}\Rightarrow {\log }_{a}x=\frac{{\log }_a{2}^{-1}}{{\log }_{a}5}={\log }_5{2}^{-1}$
$\Rightarrow x=a^{{\log }_5{2}^{-1}}=2^{-{\log }_{5}a}$