Apr 7, 2017

$- \infty < x < 1$

Explanation:

Solve the function for $y$: $y = {\log}_{a} \left(a - {a}^{x}\right)$.

Since the logarithm can only accept values greater than $0$, $a - {a}^{x} > 0$, or ${a}^{x} < a$. We can take the natural logarithm of both sides to get $x \ln \left(a\right) < \ln \left(a\right)$.

Since $a > 1$, $\ln \left(a\right) > 0$. We can divide both sides by $\ln \left(a\right)$ without worrying about flipping the inequality sign or dividing by $0$. Therefore, $x \ln \left(a\right) < \ln \left(a\right)$ becomes $x < 1$.

Thus, the domain is $- \infty < x < 1$.