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Logarithmic Differentiation

Let, fx=log...

Question

Let, $f (x) = {log}_{2} {log}_{3} {log}_{4} {log}_{5} (sin x + a^{2})$ . Find the values of $a$ for which the domain of $f (x)$ is $R$ .

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Solution

To make domain R,all logA must be valid,therefore,
$sin x + a^{2} \geq 5^{4}$
$sin x + a^{2} \geq 625$
$sin x \geq 625 - a^{2}$
$625 - a^{2} \leq sin x$
$S o, 625 - a^{2} \leq - 1$
$a^{2} \geq 625 + 1$
$a^{2} \geq 626$
$- \sqrt{626} \leq a \leq \sqrt{626}$

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