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Fundamental Laws of Logarithms

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Question

The solution of the equation ${log}_{7} {log}_{5} (\sqrt{x + 5} + \sqrt{x}) = 0$ is....

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Solution

${log}_{7} {log}_{5} (\sqrt{x + 5} + \sqrt{x}) = 0$
$\Rightarrow {log}_{5} (\sqrt{x + 5} + \sqrt{x}) = 1$
$\Rightarrow \sqrt{x + 5} + \sqrt{x} = 5 \Rightarrow \sqrt{x + 5} = 5 - \sqrt{x}$
Squaring both the sides,
$x + 5 = 25 + x - 10 \sqrt{x} \Rightarrow \sqrt{x} = 2 ∴ x = 4$

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