1. **State the problem:** Solve the equation $3^{1-2x} = 4^x$ for $x$. 2. **Recall the formula and rules:** When bases are different, take the logarithm of both sides to solve for the variable. 3. Take the natural logarithm (ln) of both sides: $$\ln(3^{1-2x}) = \ln(4^x)$$ 4. Use the logarithm power rule $\ln(a^b) = b \ln(a)$: $$ (1-2x) \ln(3) = x \ln(4) $$ 5. Distribute $\ln(3)$: $$ \ln(3) - 2x \ln(3) = x \ln(4) $$ 6. Group terms with $x$ on one side: $$ \ln(3) = x \ln(4) + 2x \ln(3) = x(\ln(4) + 2 \ln(3)) $$ 7. Solve for $x$: $$ x = \frac{\ln(3)}{\ln(4) + 2 \ln(3)} $$ 8. This is the exact solution. You can approximate it using a calculator if needed.