1. **State the problem:** Simplify the expression $$\frac{\log 10^{-4}}{2 \cdot \log_c \sqrt[3]{c}}$$. 2. **Recall logarithm properties:** - $$\log a^b = b \log a$$ - $$\log_c c = 1$$ - $$\sqrt[3]{c} = c^{\frac{1}{3}}$$ 3. **Simplify numerator:** $$\log 10^{-4} = -4 \log 10$$ Since $$\log 10 = 1$$ (common log base 10), $$\log 10^{-4} = -4$$ 4. **Simplify denominator:** $$2 \cdot \log_c \sqrt[3]{c} = 2 \cdot \log_c c^{\frac{1}{3}} = 2 \cdot \frac{1}{3} \log_c c = 2 \cdot \frac{1}{3} \cdot 1 = \frac{2}{3}$$ 5. **Combine numerator and denominator:** $$\frac{-4}{\frac{2}{3}} = -4 \times \frac{3}{2} = -6$$ **Final answer:** $$-6$$