Is $\exp(x)$ the same as $e^x$?

Yes. They are the same thing.

When exponents get really really complicated, mathematicians tend to start using $\exp(\mathrm{stuff})$ instead of $e^{\mathrm{stuff}}$.

For example: $e^{x^5+2^x-7}$ is kind of hard to read. So instead one might write: $\exp(x^5+2^x-7)$.

Note: For those who use Maple or other computer algebra systems, e^x is not usually the same as exp(x). In Maple, e^x is the variable $e$ raised to the variable $x$ power whereas exp(x) is Euler's number $e$ raised to the $x$ power.

Yes. The purpose for the notation $\exp$ is twofold:

• It allows one to talk about the exponentiation function itself, without specifying a particular input. For example, one can write that $\exp$ is a homomorphism from the additive group on $\mathbb{R}$ to the multiplicative group on $\mathbb{R}$. One may also say that $\exp$ and $\log$ are inverses.

• It allows you to write exponentiation without pushing the body of exponentiation into a superscript. For example, one may write the following, which is unwieldy to write without $\exp$ notation:

$$\prod_i e^{x_i} = \exp \sum_i x_i$$

As other answers say, in your homework (and, indeed, in most places in mathematics) there is no difference.

I have seen a beginning textbook first defining a certain function $\exp(x)$, then proving certain properties of it, and finally using those properties to motivate calling it $e^x$.