For which value of $e^x$ there is no $x$?

Question

The function $y=f(x)=e^x$ is often called injective but non-surjective. It is easy to understand why is it injective. But I don't understand why is it non-surjective? For which value of $e^x$ there is no $x$?

Answer 1

To discuss about surjective, we should specify the domain and codomain.

$$f: \mathbb{R} \rightarrow \mathbb{R}$$ $$f(x)=\exp(x)$$ is not a surjection as it doesn't take negative value.

$$f: \mathbb{R} \rightarrow \{x| x>0, x \in \mathbb{R}\}$$

$$f(x)=\exp(x)$$ is a surjection as every positive number has a preimage.

Answer 2

The question you should have asked is: for which value $y$ there is no $x$, such that $y = e^x$. The answer: for all $y\leq 0$.