1. **State the problem:** Solve the equation $$19^{4 - 2x} - 8 = 38$$ and round the solution to the nearest ten-thousand. 2. **Isolate the exponential term:** Add 8 to both sides to get $$19^{4 - 2x} = 38 + 8$$ $$19^{4 - 2x} = 46$$ 3. **Take the logarithm of both sides:** Use the natural logarithm (ln) for convenience: $$\ln\left(19^{4 - 2x}\right) = \ln(46)$$ 4. **Apply the logarithm power rule:** $$ (4 - 2x) \ln(19) = \ln(46) $$ 5. **Solve for $x$:** $$ 4 - 2x = \frac{\ln(46)}{\ln(19)} $$ $$ -2x = \frac{\ln(46)}{\ln(19)} - 4 $$ $$ x = \frac{4 - \frac{\ln(46)}{\ln(19)}}{2} $$ 6. **Calculate the numerical value:** Calculate the logarithms: $$ \ln(46) \approx 3.8286 $$ $$ \ln(19) \approx 2.9444 $$ Then, $$ \frac{\ln(46)}{\ln(19)} \approx \frac{3.8286}{2.9444} \approx 1.3001 $$ Substitute back: $$ x = \frac{4 - 1.3001}{2} = \frac{2.6999}{2} = 1.34995 $$ 7. **Round to the nearest ten-thousand:** $$ x \approx 1.3500 $$ **Final answer:** $$ \boxed{1.3500} $$