Question

The number of integral value(s) of $a$ for which ${\log }_e(x^2+5x)={\log }_e(x+a+3)$ has exactly one solution is

• A. $4$
• B. $1$
• C. $2$
• D. $5$

Answer 1

The correct option is D $5$

${\log }_e(x^2+5x)={\log }_e(x+a+3)$
For the $\log$ to be defined,
$x^2+5x>0x+a+3>0$
Drawing the graph, we get


Now, ${\log }_e(x^2+5x)={\log }_e(x+a+3)$ has $1$ solution when $x^2+5x$ and $x+a+3$ intersect at exactly one point, which is possible when
$-5\le -(a+3)<0\Rightarrow 5\ge a+3>0\Rightarrow 2\ge a>-3\therefore a=-2,-1,0,1,2$