<s>[INST] 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.
 [/INST] $$
(\sqrt{2+x})^3\sqrt{2-x}=125\sqrt{4-x^2}\\
(\sqrt{2+x})^3\sqrt{2-x}=125\sqrt{2-x}\sqrt{2+x}\\
(\sqrt{2+x})^2=125\\
2+x=125\\
x=123.
$$
Note that you have the following constraints on $x$:
$$
\sqrt{2+x}>0\implies 2+x>0\implies x>-2\\
\sqrt{2-x}>0\implies 2-x>0\implies x<2
$$
If the original equation has a solution, it should be a value between the numbers $-2$ and $2$. The answer $123$ that we found does not belong to that interval. So, it looks like the original equation does not have solutions in real numbers.
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