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  4. The solution of the equation log7 log5 ( √ x + 5 + √ x) = 0 is ....

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Q. The solution of the equation
$ log_7 \, log_5 ( \sqrt{ x + 5 } + \sqrt x) = 0 $ is ....

Solution:

Given equation is
$ log_7 \, log_5 ( \sqrt{ x + 5 } + \sqrt x) = 0 $
$\Rightarrow log_5 ( \sqrt x + 5 + \sqrt x) = 1 $
$\Rightarrow \sqrt { x + 5} + \sqrt x = 5 $
$\Rightarrow \sqrt { x + 5 } = 5 - \sqrt x $
$\Rightarrow x + 5 = 25 + x - 10 \sqrt x $
$\Rightarrow 10 \sqrt x = 20 $
$\Rightarrow \sqrt x = 2 $
$\Rightarrow x = 4$