1. **State the problem:** Solve the exponential equation $$128^x = 32$$ by expressing each side as a power of the same base and then equating exponents. 2. **Identify the bases:** Both 128 and 32 are powers of 2. Specifically, $$128 = 2^7$$ and $$32 = 2^5$$. 3. **Rewrite the equation:** Substitute these powers of 2 into the equation: $$ (2^7)^x = 2^5 $$ 4. **Simplify the left side:** Use the power of a power rule $$ (a^m)^n = a^{mn} $$: $$ 2^{7x} = 2^5 $$ 5. **Equate the exponents:** Since the bases are the same and nonzero, the exponents must be equal: $$ 7x = 5 $$ 6. **Solve for $$x$$:** $$ x = \frac{5}{7} $$ 7. **Write the solution set:** $$ \{ \frac{5}{7} \} $$ **Final answer:** The solution set is $$\left\{ \frac{5}{7} \right\}$$.