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Summation by Sigma Method

Solve the fol...

Question

Solve the following inequality:
${log}_{5} (x - 3) + \frac{1}{2} {log}_{5} 3 < \frac{1}{2} {log}_{5} (2 x^{2} - 6 x + 7)$

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Solution

$2 {log}_{5} (x - 3) + {log}_{5} 3 < {log}_{5} (2 x^{2} - 6 x + 7) {log}_{5} 3 {(x - 3)}^{2} < {log}_{5} (2 x^{2} - 6 x + 7) 3 {(x - 3)}^{2} < (2 x^{2} - 6 x + 7) 3 x^{2} - 18 x + 27 < 2 x^{2} - 6 x + 7 x^{2} - 12 x + 20 < 0 (x - 2) (x - 10) < 0 x \in (2, 10)$
Also,
$x - 3 > 0 x > 3$ and $2 x^{2} - 6 x + 7 > 0 x \in R$
Taking intersection, we get $x \in (3, 10)$

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