Final answer:
To find the domain of the function f(x) = ln(e^x - 9), the argument of the natural logarithm must be set greater than zero. After solving the inequality e^x > 9 by taking the natural logarithm of both sides, the domain in interval notation is (ln(9), ∞).
Step-by-step explanation:
The domain of the function f(x) = ln(ex - 9) is the set of all real numbers x for which the argument of the natural logarithm is positive. Because the natural logarithm function is only defined for positive arguments, we set the inside of the logarithm greater than zero.
ex - 9 > 0
ex > 9
By taking the natural logarithm of both sides, which is the inverse operation of the exponential function, we get:
x > ln(9)
Therefore, the domain of f(x) in interval notation is (ln(9), ∞).