1. **State the problem:** Solve the exponential equation $$37^{-4x} + 16 = 41$$ for $x$ and approximate the result to three decimal places. 2. **Isolate the exponential term:** Subtract 16 from both sides: $$37^{-4x} = 41 - 16$$ $$37^{-4x} = 25$$ 3. **Rewrite the equation:** The equation is now: $$37^{-4x} = 25$$ 4. **Take the natural logarithm of both sides:** $$\ln\left(37^{-4x}\right) = \ln(25)$$ 5. **Use the logarithm power rule:** $$-4x \ln(37) = \ln(25)$$ 6. **Solve for $x$:** $$x = \frac{-\ln(25)}{4 \ln(37)}$$ 7. **Calculate the values:** $$\ln(25) \approx 3.2189$$ $$\ln(37) \approx 3.6109$$ 8. **Substitute and simplify:** $$x = \frac{-3.2189}{4 \times 3.6109} = \frac{-3.2189}{14.4436} \approx -0.223$$ **Final answer:** $$x \approx -0.223$$