Question

The number of solution (s) of the equation $\frac{1}{2}{\log }_{\sqrt{3}}\left(\frac{x+1}{x+5}\right)+{\log }_9(x+5{)}^2=1$ are/is

Answer 1

$\frac{1}{2}lo{g}_{\sqrt{3}}\left(\frac{x+1}{x+5}\right)+lo{g}_9(x+5{)}^2=1$

$\Rightarrow \frac{1}{2}lo{g}_{{\sqrt{3}}^{1/2}}\left(\frac{x+1}{x+5}\right)+2lo{g}_{3^2}(x+5{)}^2=1$

$\Rightarrow 2\times \frac{1}{2}lo{g}_3\left(\frac{x+1}{x+5}\right)+2\times \frac{1}{2}lo{g}_3(x+5)=1$

$\Rightarrow lo{g}_3\left(\frac{x+1}{x+5}\right)+lo{g}_3(x+5)=1$

$\Rightarrow lo{g}_3[\left(\frac{x+1}{x+5}\right)+(x+5\left)\right]=1$ {$loga+logb=logab$}

$\Rightarrow (x+1)=3$

$\Rightarrow x+1=3$

$\Rightarrow x=2$

$\therefore$ Number od solution = 1