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

Question

For homework I have to find the derivative of $\text {exp}(6x^5+4x^3)$ but I am not sure if this is equivalent to $e^{6x^5+4x^3}$ If there is a difference, what do I do to calculate the derivative of it?

Answer 1

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.

Answer 2

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:

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

Answer 3

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 $e^x$.

Answer 4

I agree with these two answers, but I want to add one thing: Well defines.

$e$ is some (positive) number, so (without knowing the function $\exp$), you can compute $e^n$ for $n \in \mathbb{N}$ – just multiply $e$ $n$ times with itself. You can also compute $e^{-n} = \frac{1}{e^n}$ and even $e^\frac{p}{q} = \sqrt[q] e^p$ (for $n, q \in \mathbb{N}, p \in \mathbb{Z}$). One can prove that the $\exp$ function yields the same numbers with these arguments. This justifies the notation.

Answer 5

While both expressions are generally the same, $\exp(x)$ is well-defined for a really large slurry of argument domains via its series: $x$ can be complex, imaginary, or even quadratic matrices.

The basic operation of exponentiation implicated by writing $e^x$ tends to have ickier definitions, like having to think about branches when writing $e^{1\over2}$ or at least generally $a^b$.

Exponentiation can be replaced by using $\exp$ and $\ln$ together via $a^b=\exp(b\ln a)$, and the ambiguities arise from the $\ln$ part of the replacement. So it can be expedient to work with just $\exp$ when the task does not require anything else. Informally, $e^x$ is used equivalently to $\exp(x)$ anyway but the latter is really more fundamental and well-defined.

Answer 6

Although I agree with the answers already provided that in this situation (and indeed in most other ones in mathematics) there is no difference between the two notations, I would like to add the following for completeness:

In manifold theory (most particularly Lie Group theory or Riemannian geometry), the exponential map $\exp$ is a map from a tangent space to the manifold itself. For Lie groups, it expresses the local group structure and allows to lift many problems from the group to the tangent space (the Lie algebra). It also defines integral curves on the manifold and is therefore related to geodesics (which is more obvious from the viewpoint of Riemannian geometry).

This exponential $\exp$ coincides with the usual exponential for the case of the Lie group $\mathbb{R}$. It also coincides with the definition of the matrix exponential $$ e^A = \sum_{n=0}^\infty\frac{A^n}{n!}. $$ However, I believe this cannot be done in general, although I do not have an example available.

Answer 7

Another reason we use $\exp(x)$ is when defining it in terms of its power series. At that point, we don't know that $\exp(x)=e^x$ when $x$ is real.

Also, there are general problems using the notation $a^z$ when $z$ is complex. $a^z$ is actually a multi-valued function. $e^z$ is thus sometimes ambiguous, so we will, in those cases, prefer $\exp(z)$ to clarify that we are talking about the single-valued function.

Answer 8

We use exp(Θ) for discrete values. Matlab can only use exp() because everything is discrete in it. In reality, you can use e^Θ in your algebra because continuous.

You can still consider (exp^(jΘ)-e^(jΘ)) / 2j to be a sin if you want discrete values.