How do I solve this log equation for x? $3\log _5\left(\sqrt{2+x}\right)+\log _5\left(\sqrt{2-x}\right)-3=\log _5\left(\sqrt{4-x^2}\right)$ $3\log _5\left(\sqrt{2+x}\right)+\log _5\left(\sqrt{2-x}\right)-3=\log _5\left(\sqrt{4-x^2}\right)$
I only managed to solve up to this step: $\left(2+x\right)^{\frac{3}{2}}\sqrt{2-x}=\sqrt{4-x^2}\cdot \:125$
However, I'm not too sure about squaring both sides as I would have to handle ridiculously large numbers to find x, so I'm not too sure that I'm doing it correctly.

You are almost there. 
$$\left(2+x\right)^{\frac{3}{2}}\sqrt{2-x}=\sqrt{4-x^2}\cdot125$$
$$\implies (2+x)\sqrt{2+x}\cdot\sqrt{2-x} = 125\sqrt{(2+x)(2-x)}$$
Here, notice that $x = 2$ and $x = -2$ are not the solutions for the original equation because $\log_50$ is not defined. Therefore, we can divide both sides by $\sqrt{2+x}\sqrt{2-x}$. Then, we have
$$2+x = 125 \implies x = 123$$
But this solution makes $\sqrt{2-x}$ a non-real number. Therefore this equation has no real solution. 
