1. **State the problem:** We need to find values of $x$ and $y$ such that $$3^x - 3^y = 234.$$ 2. **Rewrite the equation:** We can factor the left side using the property of exponents: $$3^x - 3^y = 3^y(3^{x-y} - 1) = 234.$$ 3. **Set variables:** Let $a = y$ and $b = x - y$. Then the equation becomes $$3^a(3^b - 1) = 234.$$ 4. **Analyze factors:** Since $3^a$ and $3^b - 1$ are integers, we look for integer values of $a$ and $b$ such that their product is 234. 5. **Factor 234:** The prime factorization of 234 is $$234 = 2 \times 3^2 \times 13.$$ 6. **Try possible values for $a$:** Since $3^a$ divides 234, possible values for $a$ are 0, 1, or 2 because $3^3=27$ and $27 \times (3^b -1)$ would exceed 234 for small $b$. - For $a=0$: $$3^0 = 1,$$ so $$3^b - 1 = 234 \Rightarrow 3^b = 235,$$ which is not a power of 3. - For $a=1$: $$3^1 = 3,$$ so $$3^b - 1 = \frac{234}{3} = 78 \Rightarrow 3^b = 79,$$ not a power of 3. - For $a=2$: $$3^2 = 9,$$ so $$3^b - 1 = \frac{234}{9} = 26 \Rightarrow 3^b = 27,$$ which is $$3^3.$$ 7. **Find $b$:** From above, $$3^b = 27 = 3^3 \Rightarrow b = 3.$$ 8. **Recall definitions:** $$a = y = 2,$$ $$b = x - y = 3 \Rightarrow x = b + y = 3 + 2 = 5.$$ 9. **Final answer:** $$x = 5, \quad y = 2.$$