The set of values of $x$ for which logarithm expression
${log}_{2} (x^{2} - x - 6) + {log}_{0.5} (x - 3)$
to be defined is

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Solution

Given :
${log}_{2} (x^{2} - x - 6) + {log}_{0.5} (x - 3)$

For expression to be defined
$x^{2} - x - 6 > 0 and x - 3 > 0$

$\Rightarrow x^{2} - x - 6 > 0 and x - 3 > 0$

$\Rightarrow x^{2} - 3 x + 2 x - 6 > 0 and x - 3 > 0$

$\Rightarrow (x - 3) (x + 2) > 0 and x - 3 > 0$

Common port between these two conditions $x \in (3, \infty)$

So, above given logarithmic expression is defined only in the interval of $x \in (3, \infty)$

Hence the correct answer is Option B.

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