Question

The solution set of ${\log }_5(x+20){\log }_x\left(\sqrt{5}\right)=1$ is

• A. $\left\{25\right\}$
• B. $\varphi$
• C. $\left\{5\right\}$
• D. $\{5,25\}$

Answer 1

Answer: (C)

$\left\{5\right\}$

Equation ${\log }_5(x+20){\log }_x\left(\sqrt{5}\right)=1$ is valid, when $x>0$ & $x\ne 1$

$\Rightarrow {\log }_5(x+20)={\log }_{\sqrt{5}}x[\because {\log }_{a}b=\frac{1}{{\log }_{b}a}]$
$\Rightarrow {\log }_5(x+20)=2{\log }_{5}x={\log }_5{x}^2[\because m\log a=\log a^m]$
$\Rightarrow x^2-x-20=0$
$\Rightarrow x=5,-4$
Since, $x>0$
Therefore, $x=5$
Ans: C