Question

Sum of all the values of $x$ satisfying the equation ${\log }_{17}{\log }_{11}(\sqrt{x+11}+\sqrt{x})=0$ is:

• A. 25
• B. 36
• C. 171
• D. 0

Answer 1

Answer: (A)

Equation (1) is defined if $x\ge 0$.
We can rewrite (1) as
${\log }_{11}(\sqrt{x+11}+\sqrt{x})=17^0=1$
$\Rightarrow \sqrt{x+11}+\sqrt{x}=11^1=11$
$\Rightarrow \sqrt{x+11}=11-\sqrt{x}$
Squaring both the sides we get
$x+11=121-22\sqrt{x}+x$
$\Rightarrow 22\sqrt{x}=110\Rightarrow \sqrt{x}=5\text{ or }x=25$
This clearly satisfies (1).
Thus, sum of all the values satisfying (1) is 25 .