1. **State the problem:** Solve for $x$ in the equation $9 \cdot 2^{2x} = 1$ and round the answer to the nearest hundredth. 2. **Rewrite the equation:** The equation is $9 \times 2^{2x} = 1$. 3. **Isolate the exponential term:** Divide both sides by 9: $$\frac{9 \times 2^{2x}}{9} = \frac{1}{9}$$ which simplifies to $$\cancel{9} \times 2^{2x} / \cancel{9} = \frac{1}{9}$$ so $$2^{2x} = \frac{1}{9}$$ 4. **Take the logarithm of both sides:** Use natural logarithm (ln) or log base 2. Using natural logarithm: $$\ln\left(2^{2x}\right) = \ln\left(\frac{1}{9}\right)$$ 5. **Use logarithm power rule:** $$2x \ln(2) = \ln\left(\frac{1}{9}\right)$$ 6. **Solve for $x$:** $$x = \frac{\ln\left(\frac{1}{9}\right)}{2 \ln(2)}$$ 7. **Calculate the values:** $$\ln\left(\frac{1}{9}\right) = \ln(1) - \ln(9) = 0 - \ln(9) = -\ln(9)$$ $$\ln(9) \approx 2.197224577$$ $$\ln(2) \approx 0.693147181$$ 8. **Substitute and compute:** $$x = \frac{-2.197224577}{2 \times 0.693147181} = \frac{-2.197224577}{1.386294362} \approx -1.585$$ 9. **Round to nearest hundredth:** $$x \approx -1.59$$ **Final answer:** $x \approx -1.59$