Finding the domain of logarithmic function

Question

I got confused trying to find out the domain of the logarithmic function below: $$f(x)=\ln(e^x+3)$$ Because the argument of $f,\,e^x+3,$ is a nonnegative number, $e^x+3>0$ and $e^x>-3.$ Taking natural logarithms on both sides, we get $x>\ln(-3)$. However, the domain of a logarithmic function must be nonnegative real numbers, so $\ln(-3)$ doesn't make sense. Then how can the domain of the original function be defined?

Answer 1

The domain in interval notation is $(-\infty,\infty)$, or $x\in\mathbb{R}$.

You are trying to find out when $e^x+3\le 0$, so you can state restrictions in the domain.

$e^x\le-3\implies x\le\ln (-3)$

Since $\ln (-3)$ is not defined for real numbers, there are no restrictions.

Answer 2

$e^x+3>0$ for all $x\in\mathbb{R}$, so $\log(e^x+3)$ is defined for all $x\in\mathbb{R}$