1. **State the problem:** Solve for $x$ in the equation $$3^x + 2^x = 13.$$\n\n2. **Understand the equation:** We want to find the value of $x$ such that the sum of $3^x$ and $2^x$ equals 13. Both terms are exponential functions with different bases.\n\n3. **Try integer values:** Since $3^x$ and $2^x$ grow quickly, test some integer values of $x$.\n\n4. **Check $x=2$:** $$3^2 + 2^2 = 9 + 4 = 13.$$ This satisfies the equation exactly.\n\n5. **Verify uniqueness:** Both $3^x$ and $2^x$ are strictly increasing functions, so the sum is also strictly increasing. Therefore, there is only one solution.\n\n**Final answer:** $$x = 2.$$