# Solve for: $8\log_4\sqrt{x^2-9}+3\sqrt{2\log_4\left(x+3\right)^2}=10+\log_2\left(x-3\right)^2$

Solve for: $$8\log_4\sqrt{x^2-9}+3\sqrt{2\log_4\left(x+3\right)^2}=10+\log_2\left(x-3\right)^2$$

My try:

$$8\log_4\sqrt{x^2-9}+3\sqrt{2\log_4\left(x+3\right)^2}=10+\log_2\left(x-3\right)^2\\\Leftrightarrow \log_2\left(x^2-9\right)^2+3\sqrt{\log_2\left(x+3\right)^2}=10+\log_2\left(x-3\right)^2\\\Leftrightarrow \log_2\left(x-3\right)^2+\log_2\left(x+3\right)^2+3\sqrt{\log_2\left(x+3\right)^2}-10-\log_2\left(x-3\right)^2=0\\\Leftrightarrow \log_2\left(x+3\right)^2+3\sqrt{\log_2\left(x+3\right)^2}-10=0$$

But I don't know Conditions defined for this math? Could you help me please?

$$\log_2(x+3)=y$$ $$y^2+3y-10=0$$ $$y_{1,2}=\frac{-3\pm7}{2},y_1=2,y_2=-5$$ $$\log_2(x+3)=2,x+3=4,x=1$$ $$\log_2(x+3)=-5,x+3=2^{-5},x=2^{-5}-3$$