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What is the easiest way to solve the following equation for g in terms of x? I seem to be going in circles, and don't know how exactly to deal with the $\pm$ symbol.

Equation: $$\pm \sqrt{2}x + \ln(2\sqrt{6}+1)= \ln\left|\frac{\sqrt{g+2}-\sqrt{3}}{\sqrt{g+2}+\sqrt{3}}\right|$$

I would really appreciate some help!

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Just take it apart step by step: first exponentiate both sides: $$e^{\rm LHS}=\frac{\sqrt{g+2}-\sqrt{3}}{\sqrt{g+2}+\sqrt{3}}\ .$$ Multiply out, $$e^{\rm LHS}(\sqrt{g+2}+\sqrt{3})=\sqrt{g+2}-\sqrt{3}\ ,$$ expand, now you should easily be able to find $\sqrt{g+2}$, then square and subtract $2$.

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