Number of integral solutions of inequality ((1)/(10)) ^ _ (x - 3)...

${\left(\dfrac{1}{10}\right)}^{\log_{\left(x - 3\right)} \left(x^{2} - 4x + 3\right)} \ge {\left(\dfrac{1}{10}\right)}^{0} ⇒ \log_{\left(x - 3\right)} \left(x^{2} - 4x + 3\right) \le 0$ ~ ~ ~ & (base is less than 1)

Case - I : \left(x - 3\right) > 1$and$0 < x^{2} - 4x + 3 \le 1$<br /><br />$⇒ x > 4$and$( x - 1 ) ( x - 3 ) > 0$and$x^{2} - 4x + 2 \le 0$<br /><br />$⇒ x > 4$and$x^{2} - 4x + 2 \le 0 ⇒ x > 4$and$2 - \sqrt{2} \le x \le 2 + \sqrt{2} ∴$not possible<br /><br />$∴$~ No Solution<br /><br /><strong>Case - II :</strong>$0 < \left(x - 3\right) < 1$and$x^{2} - 4x + 3 \ge 1$<br /><br />$⇒ 3 < x < 4$and$x \le 2 - \sqrt{2}x \ge 2 + \sqrt{2}$<br /><br />$⇒ 2 + \sqrt{2} \le x < 4 ⇒$there is no integral value of x.$