[tex]\displaystyle \sf \log_{2^{-x}} \left(-\sqrt[3]{\sf x^2+2x-3}\right) > \log_{2^{-x}} \left(\sqrt[3]{3}\right) \\\\ \underline{\text{Condi\c c\~ao de Exist\^encia (C.E)}} : \\\\ (1) \ \\\\ -\sqrt[3]{\sf x^2+2x-3} > 0 \to x^2+2x-3 < 0 \\\\ (x-1)(x+3) < 0 \\\\ -3 < x < 1 \\\\ (2)\\\\ 1\neq 2^{-x} > 0 \\\\ 2^{-x} > 0 \to x\in\mathbb{R} \ \text{(exponencial \'e positiva )} \\\\ 2^{-x} \neq 1 \to x \neq 0 \\\\\ (1) \cap(2) : \\\\ \boxed{\sf C.E : \ -3 < x < 1 \ e \ x\neq 0 }[/tex]

vamos melhorar a equação :

[tex]\displaystyle \sf \log_{2^{-x}} \left(-\sqrt[3]{\sf x^2+2x-3}\right) > \log_{2^{-x}} \left(\sqrt[3]{3}\right) \\\\\\ \log_{\displaystyle \left(\frac{1}{2}\right)^{x}} \left(\sqrt[3]{\sf -x^2-2x+3}\right) > \log_{\displaystyle \left(\frac{1}{2}\right)^{x}}\left(\sqrt[3]{3}\right) \\\\\\ \log_{\displaystyle \left[\left(\frac{1}{2}\right)^{x} \right]} \left(-x^2-2x+3 \right)^{\frac{1}{3}} > \log_{\displaystyle \left[\left(\frac{1}{2}\right)^{x} \right]}\left(3\right)^{\frac{1}{3}} \\\\\\[/tex]

[tex]\displaystyle \sf \frac{1}{3} \cdot \log_{\displaystyle \left[\left(\frac{1}{2}\right)^{x} \right]} \left(-x^2-2x+3 \right) > \frac{1}{3}\cdot \log_{\displaystyle \left[\left(\frac{1}{2}\right)^{x} \right]}\left(3\right) \\\\\\ \log_{\displaystyle \left[\left(\frac{1}{2}\right)^{x} \right]} \left(-x^2-2x+3 \right) > \log_{\displaystyle \left[\left(\frac{1}{2}\right)^{x} \right]}\left(3\right)[/tex]

1º caso :

[tex]\displaystyle \sf 2^{-x} > 1 \to \left(\frac{1}{2}\right)^{x} > 1 \to x < 0 \\\\\\ Da{\'i}}: \\\\ \log_{\displaystyle \left[\left(\frac{1}{2}\right)^{x} \right]} \left(-x^2-2x+3 \right) > \log_{\displaystyle \left[\left(\frac{1}{2}\right)^{x} \right]}\left(3\right) \\\\\\\ -x^2-2x+3 > 3 \\\\ -x^2-2x > 0 \to x^2+2x < 0 \\\\\ x(x+2) < 0 \to \boxed{\sf -2 < x < 0}[/tex]

2º caso :

[tex]\displaystyle \sf \left(\frac{1}{2}\right)^{x} < 1 \to x > 0 \\\\\\ Da{\'i}}: \\\\ \log_{\displaystyle \left[\left(\frac{1}{2}\right)^{x} \right]} \left(-x^2-2x+3 \right) > \log_{\displaystyle \left[\left(\frac{1}{2}\right)^{x} \right]}\left(3\right) \\\\\\\ -x^2-2x+3 < 3 \\\\ -x^2-2x < 0 \to x^2+2x > 0 \\\\\ x(x+2) > 0 \to \boxed{\sf x < -2 \ \ ou\ \ x > 0}[/tex]

Fazendo a interseção dos três intervalos :

[tex]\displaystyle \sf C.E\cap (1^\circ caso)\cap (2^\circ \ caso) \\\\ (-3 < x < 1) \cap (x\neq 0 ) \cap(-2 < x < 0) \cap (x < -2 ) \cap ( x > 0) \\\\\\ \Large \boxed{\sf \begin{matrix} \ \text{Conjunto Solu\c c\~ao} : \\\\ \sf (-2,0)\ \cup \ (0,1) \\ _ \end{matrix}\ }\checkmark[/tex]