1. **State the problem:** Solve the equation $$7^x = 3^x + 4$$ for $x$. 2. **Understand the equation:** We want to find the value of $x$ such that the exponential expression on the left equals the sum on the right. 3. **Rewrite the equation:** $$7^x - 3^x = 4$$ 4. **Consider the behavior of the function:** Define $$f(x) = 7^x - 3^x - 4$$. We want to find $x$ such that $f(x) = 0$. 5. **Check integer values:** - For $x=1$: $$7^1 - 3^1 - 4 = 7 - 3 - 4 = 0$$ 6. **Conclusion:** Since $f(1) = 0$, $x=1$ is a solution. 7. **Check if there are other solutions:** - For $x=0$: $$7^0 - 3^0 - 4 = 1 - 1 - 4 = -4 < 0$$ - For $x=2$: $$7^2 - 3^2 - 4 = 49 - 9 - 4 = 36 > 0$$ Since $f(x)$ is continuous and increasing (because $7^x$ grows faster than $3^x$), and it crosses zero at $x=1$, this is the unique solution. **Final answer:** $$x = 1$$