ĐKXĐ: \(0< x\le1\)
\(\Leftrightarrow\log\left(1+\sqrt{1-x^2}\right)=2\log\left(2+\sqrt{1-x^2}\right)-\log\left(\sqrt{\left(1+x\right)^3}-\sqrt{\left(1-x\right)^3}\right)^2\)
\(\Leftrightarrow\log\left(1+\sqrt{1-x^2}\right)=2\log\left(2+\sqrt{1-x^2}\right)-\log\left(6x^2+2-2\sqrt{\left(1-x^2\right)^3}\right)\)
Đặt \(\sqrt{1-x^2}=t\Rightarrow0\le t< 1\)
\(\Rightarrow\log\left(1+t\right)=2\log\left(2+t\right)-\log\left(8-6t^2-2t^3\right)\)
\(\Leftrightarrow\log\left(1+t\right)=2\log\left(t+2\right)-\log\left[\left(2-2t\right)\left(t+2\right)^2\right]\)
\(\Leftrightarrow\log\left(1+t\right)=2\log\left(t+2\right)-\log\left(2-2t\right)-2\log\left(t+2\right)\)
\(\Leftrightarrow\log\left(1+t\right)+\log\left(2-2t\right)=0\)
\(\Leftrightarrow\left(1+t\right)\left(2-2t\right)=1\)
\(\Leftrightarrow t^2=\frac{1}{2}\)
\(\Leftrightarrow1-x^2=\frac{1}{2}\)
