The least value of |z| where z is complex number which satisfies...

{e}^{\left(\dfrac{\left(\left|z\right| + 3\right)\left(\left|z\right| - 1\right)}{\left|\left|z\right| + 1\right|}\log_{e}2\right)} \ge \log_{\sqrt{2}}\left|5\sqrt{7} + 9i\right|$</p><p>$⇒ 2\dfrac{\left(\left|z\right| + 3\right)\left(\left|z\right| - 1\right)}{\left(\left|z\right| + 1\right)} \ge lo{g}_{\sqrt{2}}\left(16\right)$</p><p>$⇒ 2\dfrac{\left(\left|z\right| + 3\right)\left(\left|z\right| - 1\right)}{\left(\left|z\right| + 1\right)} \ge {2}^{3}$</p><p>$\dfrac{\left(\left|z\right| + 3\right)\left(\left|z\right| - 1\right)}{\left(\left|z\right| + 1\right)} \ge 3$</p><p>\left(\left|z\right| + 3\right)\left(\left|z\right| - 1\right) \ge 3\left(\left|z\right| + 1\right)$

${\left|z\right|}^{2} + 2\left|z\right| - 3 \ge 3\left|z\right| + 3$

$⇒ {\left|z\right|}^{2} + \left|z\right| - 6 \ge 0$

$⇒ \left(\left|z\right| - 3\right)\left(\left|z\right| + 2\right) \ge 0 ⇒ \left|z\right| - 3 \ge 0$

⇒ \left|z\right| \ge 3 ⇒ {\left|z\right|}_\min = 3