1. **Stating the problem:** We are given the function $f(x) = e^{ax}$ and the differential equation $f'(x) - f(x) = 0$. 2. **Recall the derivative of the function:** The derivative of $f(x) = e^{ax}$ with respect to $x$ is $f'(x) = a e^{ax}$. 3. **Substitute into the differential equation:** Replace $f'(x)$ and $f(x)$ in the equation: $$a e^{ax} - e^{ax} = 0$$ 4. **Factor the expression:** $$e^{ax}(a - 1) = 0$$ 5. **Solve for $a$:** Since $e^{ax} \neq 0$ for all real $x$, the factor $a - 1$ must be zero: $$a - 1 = 0 \implies a = 1$$ 6. **Conclusion:** The value of $a$ that satisfies the differential equation is $a = 1$. Therefore, the function is $f(x) = e^{x}$.