Question

The solution of the equation $lo{g}_7\left(lo{g}_5\left(\sqrt{x+5}+\sqrt{x}\right)\right)=0$is $x=$

Answer 1

Answer: 4

The given equation is
$lo{g}_7\left[lo{g}_5\left(\sqrt{x+5}+\sqrt{x}\right)\right]=0$
$\Rightarrow lo{g}_5\left(\sqrt{x+5}+\sqrt{x}\right)=1$
$\Rightarrow \sqrt{x+5}+\sqrt{x=5}$
$\Rightarrow \sqrt{x+5}=5-\sqrt{x}$
$\Rightarrow x+5=25-10\sqrt{x}+x$
$\Rightarrow 10\sqrt{x}=20$
$\Rightarrow \sqrt{x}=2$
$\Rightarrow x=4$
Hence, the solution is $x=4$