Question

The solution set of the equation ${\log }_{e}x+{\log }_e(x+\frac{8}{3})=0$ is

• A. $\{-3,\frac{1}{3}\}$
• B. $\{3,-\frac{1}{3}\}$
• C. $\{3,\frac{1}{3}\}$
• D. $\left\{\frac{1}{3}\right\}$

Answer 1

The correct option is D $\left\{\frac{1}{3}\right\}$

For the equality to be defined,
$x>0$ and $x+\frac{8}{3}>0$
$\Rightarrow x>0$ and $x>-\frac{8}{3}$
$\therefore x>0\cdots \left(1\right)$

Now, ${\log }_{e}x+{\log }_e(x+\frac{8}{3})=0$
$\Rightarrow {\log }_e\left[x(x+\frac{8}{3})\right]=0$
$\Rightarrow x(x+\frac{8}{3})=1$
$\Rightarrow 3x^2+8x-3=0$
$\Rightarrow (3x-1)(x+3)=0$
$\Rightarrow x=\frac{1}{3},-3$
But $x=-3$ does not satisfy $\left(1\right)$.
$\therefore x=\frac{1}{3}$