[latex]log_{2 \sqrt{2+ \sqrt{3} } }( x^{2} +2x-2)=log_{2+ \sqrt{3} }(x^2+2x-3)[/latex]

 [latex]log_{2\sqrt{2+\sqrt{3}}} (x^2+2x-2) = log_{2+\sqrt{3}} (x^2+2x-3)\\ 2*log_{ 8+4\sqrt{3}}(x^2+2x-2) = log_{ 2+\sqrt{3} } (x^2+2x-3)\\ log_{x^2+2x-3} (2+\sqrt{3})^2=log_{x^2+2x-2}(8+4\sqrt{3})\\ log_{x^2+2x-3}(7+4\sqrt{3}) = log_{x^2+2x-2}(8+4\sqrt{3})\\ x^2+2x-3=a\\ 7+4\sqrt{3}=b\\ log_{a}b=log_{a+1}(b+1)\\ a=b\\ x^2+2x-(10+4\sqrt{3})=0\\ x=-1-\sqrt{11+4\sqrt{3}}\\ x=\sqrt{11+4\sqrt{3}-1 [/latex]