Question

Solution set of the equation: ${\log }_{10}(x-9)+2{\log }_{10}\sqrt{2x-1}=2$, is

• A. $1$
• B. $13$
• C. $\frac{1}{2}$
• D. $\varphi$

Answer 1

The correct option is B $13$

${\log }_{10}(x-9)+2{\log }_{10}\sqrt{2x-1}=2$
Above equation is valid when $x-9>0$ and $2x-1>0$
$\Rightarrow x>9$
${\log }_{10}(x-9)+2{\log }_{10}\sqrt{2x-1}=2$
$\Rightarrow {\log }_{10}(x-9)+{\log }_{10}(\sqrt{2x-1}{)}^2=2,[\because m\log x=\log x^m]$
$\Rightarrow \log (x-9)(2x-1)=2,[\because \log a+\log b=\log (ab\left)\right]$
$\Rightarrow (x-9)(2x-1)=100$
$\Rightarrow x=-\frac{7}{2},13$
Since, $x>9$
Therefore, $x=13$
Ans: B