The least value of |z|, where z is a complex number satisfying ![(|z|+3)(|z|-1)|z|+12]_√2!…

Question

The least value of $|z|$, where $z$ is a complex number satisfying $\exp\!\left[\dfrac{(|z|+3)(|z|-1)}{|z|+1}\ln2\right]\ge\log_{\sqrt2}\!\left(|\sqrt7+9i|\right)$, is:

Accepted Answer

The right side evaluates to $\log_{\sqrt2}16=8$, so the inequality reduces to $\dfrac{(|z|+3)(|z|-1)}{|z|+1}\ge3$, i.e. $(|z|-3)(|z|+2)\ge0$, giving $|z|\ge3$ (as $|z|\ge0$). Least value is 3.

Solutions · 1