1. **State the problem:** Solve for $x$ in the equation $$3^x + 2^x = 13.$$\n\n2. **Understand the equation:** We have an exponential equation with two terms, $3^x$ and $2^x$, summing to 13. We want to find the value of $x$ that satisfies this.\n\n3. **Try integer values:** Since $3^x$ and $2^x$ grow quickly, test some integer values of $x$.\n\n- For $x=2$: $$3^2 + 2^2 = 9 + 4 = 13.$$\n\n4. **Check if $x=2$ satisfies the equation:** Yes, it does.\n\n5. **Conclusion:** The solution is $$x=2.$$