1. **State the problem:** Solve the exponential equation $$3^{x-1} = 2^{x+1}$$ for $x$. 2. **Use logarithms to solve:** Taking the logarithm base 3 on the left and base 2 on the right is not straightforward, so instead take the natural logarithm (ln) of both sides: $$\ln(3^{x-1}) = \ln(2^{x+1})$$ 3. **Apply logarithm power rule:** $$ (x-1) \ln 3 = (x+1) \ln 2 $$ 4. **Expand both sides:** $$ x \ln 3 - \ln 3 = x \ln 2 + \ln 2 $$ 5. **Group terms with $x$ on one side and constants on the other:** $$ x \ln 3 - x \ln 2 = \ln 3 + \ln 2 $$ 6. **Factor out $x$:** $$ x (\ln 3 - \ln 2) = \ln 3 + \ln 2 $$ 7. **Solve for $x$:** $$ x = \frac{\ln 3 + \ln 2}{\ln 3 - \ln 2} $$ 8. **Calculate the numerical value:** Using approximate values $\ln 3 \approx 1.0986$ and $\ln 2 \approx 0.6931$: $$ x = \frac{1.0986 + 0.6931}{1.0986 - 0.6931} = \frac{1.7917}{0.4055} \approx 4.419 $$ **Final answer:** $$ x \approx 4.419 $$