1. **Problem statement:** Solve for $x$ in the equation $\log_x 32 = \frac{9}{7}$. 2. **Recall the logarithm definition:** $\log_a b = c$ means $a^c = b$. 3. **Apply the definition:** From $\log_x 32 = \frac{9}{7}$, we get $$x^{\frac{9}{7}} = 32$$ 4. **Rewrite 32 as a power of 2:** $$32 = 2^5$$ 5. **Equate powers:** $$x^{\frac{9}{7}} = 2^5$$ 6. **Solve for $x$ by raising both sides to the power $\frac{7}{9}$:** $$x = \left(2^5\right)^{\frac{7}{9}} = 2^{5 \times \frac{7}{9}} = 2^{\frac{35}{9}}$$ 7. **Final answer:** $$x = 2^{\frac{35}{9}}$$