The least value of $$|z|$$ where $$z$$ is complex number which satisfies the inequality $$e^{\left(\frac{(|z|+3)(|z|-1)}{||z|+1|}\log_e 2\right)} \geq \log_{\sqrt{2}}|5\sqrt{7} + 9i|$$, $$i = \sqrt{-1}$$, is equal to:

First, we simplify the right-hand side. We compute $$|5\sqrt{7} + 9i| = \sqrt{(5\sqrt{7})^2 + 9^2} = \sqrt{175 + 81} = \sqrt{256} = 16$$.

Then $$\log_{\sqrt{2}} 16 = \frac{\ln 16}{\ln \sqrt{2}} = \frac{4 \ln 2}{\frac{1}{2} \ln 2} = 8$$.

The inequality becomes: $$2^{\frac{(|z|+3)(|z|-1)}{|z|+1}} \geq 2^3$$ (since $$e^{a \ln 2} = 2^a$$ and $$||z|+1| = |z|+1$$ because $$|z| \geq 0$$).

Since the base 2 $$>$$ 1, we can compare exponents directly: $$\frac{(|z|+3)(|z|-1)}{|z|+1} \geq 3$$.

Let $$r = |z|$$ where $$r \geq 0$$. Multiplying both sides by $$(r+1)$$, which is always positive: $$(r+3)(r-1) \geq 3(r+1)$$.

Expanding: $$r^2 + 2r - 3 \geq 3r + 3$$, which simplifies to $$r^2 - r - 6 \geq 0$$.

Factoring: $$(r-3)(r+2) \geq 0$$. Since $$r \geq 0$$, the factor $$(r+2)$$ is always positive, so we need $$r - 3 \geq 0$$, giving $$r \geq 3$$.

The least value of $$|z|$$ is $$3$$.