1. **State the problem:** Solve the equation $$5^{-8}(2^x - 1)^3 = 125$$ for $x$. 2. **Rewrite constants:** Note that $125 = 5^3$, so the equation becomes $$5^{-8}(2^x - 1)^3 = 5^3$$ 3. **Isolate the cubic term:** Multiply both sides by $5^8$ to cancel $5^{-8}$ on the left: $$\cancel{5^{-8}}(2^x - 1)^3 \times 5^8 = 5^3 \times 5^8$$ which simplifies to $$(2^x - 1)^3 = 5^{3+8} = 5^{11}$$ 4. **Take the cube root of both sides:** $$2^x - 1 = \sqrt[3]{5^{11}} = 5^{\frac{11}{3}}$$ 5. **Solve for $2^x$:** $$2^x = 1 + 5^{\frac{11}{3}}$$ 6. **Take the logarithm base 2 of both sides:** $$x = \log_2\left(1 + 5^{\frac{11}{3}}\right)$$ **Final answer:** $$x = \log_2\left(1 + 5^{\frac{11}{3}}\right)$$