1. **State the problem:** Simplify the expression $$\frac{\log\left(\frac{10}{x}\right) - 1}{\log\sqrt[5]{x}}$$. 2. **Recall logarithm properties:** - $$\log\left(\frac{a}{b}\right) = \log a - \log b$$ - $$\log a^b = b \log a$$ - The 5th root of $$x$$ is $$x^{\frac{1}{5}}$$, so $$\log\sqrt[5]{x} = \log x^{\frac{1}{5}} = \frac{1}{5} \log x$$. 3. **Apply the properties to the numerator:** $$\log\left(\frac{10}{x}\right) - 1 = \log 10 - \log x - 1$$. Since $$\log 10 = 1$$ (assuming base 10), this becomes: $$1 - \log x - 1 = -\log x$$. 4. **Rewrite the denominator:** $$\log\sqrt[5]{x} = \frac{1}{5} \log x$$. 5. **Substitute back into the expression:** $$\frac{-\log x}{\frac{1}{5} \log x}$$. 6. **Simplify the fraction:** $$= \frac{-\log x}{\frac{1}{5} \log x} = -\log x \times \frac{5}{\log x}$$. 7. **Cancel common factors:** $$= -\cancel{\log x} \times \frac{5}{\cancel{\log x}} = -5$$. **Final answer:** $$-5$$.