1. **State the problem:** We need to find the value of $x$ such that $f(x) = 1 - e^x = 0$. 2. **Set the equation to zero:** $$1 - e^x = 0$$ 3. **Isolate the exponential term:** $$e^x = 1$$ 4. **Recall the property of the exponential function:** The exponential function $e^x$ equals 1 when $x = 0$. 5. **Therefore, the solution is:** $$x = 0$$ **Final answer:** $x = 0$