Question

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

• A. $0$
• B. $1$
• C. $2$
• D. infinite

Answer 1

Answer: (B)

$1$

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

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

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

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

$\Rightarrow {\log }_3(x+1)=1\Rightarrow x+1=3^1=3$

$\Rightarrow x=2$

$\therefore$ Only one solution.