Why write $\exp(x)$, instead of $e^x$? Why write $\exp(x)$, instead of $e^x$?
I would have thought that if $x$ is a long expression, $\exp(x)$ would be preferred, so that it is clear? But I have seen that this is not the case.
 A: I use $\exp$ pretty often but I'm also often converting my stuff into programming, where $\exp(blah)$ is literally how I'd write exponentials.  It also helps when considering derivatives; converting generic exponentials $a^b$ into $\exp(b\ln a)$ makes it much more obvious what you're doing.
But on the other hand:  one of the major advantages of mathematical typesetting is that things in different contexts can have different sizes and baselines.  So in relatively complex expressions it can actually be harder to read $\exp((b+l)a)h$ as opposed to $e^{(b+l)a}h$, because you can tell when the exponent ends very easily but the parentheses can get lost in among the other parentheses.
On the other other hand, when it gets very complicated in the exponent and a lot of it isn't parentheses-based, the superscript form can get bad too.  Check out this monster excerpted from the Feynman-Kac formula:
$$e^{-\int_t^TV\left(X_\tau,\tau\right)d\tau}$$
$$\exp\left(-\int_t^TV\left(X_\tau,\tau\right)d\tau\right)$$
In this case I'd probably prefer the $\exp$ form, not least because my first reading misinterpreted the subscript on $X$ as $r$!  Thanks to @anomaly in the question comments for this particular example.
A: Yes, generally we use $\exp$ when there is a longer formula in the exponent.  No one wants to read $$e^{\sum_{k=1}^\infty \frac{1}{k^2}}$$ so we use $$\exp\left(\sum_{k=1}^\infty \frac{1}{k^2}\right)$$ instead.  Additionally, sometimes I see $\exp(x) := \sum_{k=0}^\infty \frac{x^k}{k!}$ used in complex analysis to avoid confusion with a power $e^k = e \cdot e \cdots e$.
A: I like to use $\exp(x)$, because


*

*It looks better with "big" arguments, or when the argument has an index

*I find it more elegant

*My handwriting is a bit messy, so it makes my notes easier to read


I like to use $e^x$, because


*

*It's shorter

*It's easier to use and to recognize the exponential rules in this form


But based on my (not so huge amount of) experiences, people use the $\exp$ when they define the function by its power series, i.e.
$$\exp(x):=\sum_{n \in \mathbb{N}} \frac{x^n}{n!}$$
and the $e^x$ when they approach it from the "algebraic" way, i.e. they define $a^b$ first for natural $b$, then they extend it to rationals and reals, and define $e$ as
$$e:=\lim_{n} \left(1+\frac{1}{n}\right)^n$$
A: The notation $e^x$ is to be preferred, in my opinion, because it makes it clear that the exponential function has many properties in common with the laws of indices.
For example;
$$e^{x+y}=e^x \times e^y$$ 
is easier to remember than
$$\exp(x+y)=\exp(x) \times \exp(y)$$
and
$$\frac{1}{e^x}=e^{-x}$$
is easier to remember than
$$\frac{1}{\exp(x)}=\exp(-x)$$
It's not that it's impossible to use the $\exp(x)$ form but there already enough variations of how functions with brackets expand to remember that why make it harder by having a notation that's less helpful than it could be ?
Here's an example of a bracket expansion where effort needs to be made to remember it quickly;
$$\cos(A+B)=\cos(A)\cos(B)-\sin(A)\sin(B)$$
To me, it's about focussing energy where it's needed and not squandering it with unnecessarily awkward notation.
Although, I must add, I've enjoyed reading some of the other answers that are giving reasons to switch when the index notation becomes unwieldy.
A: I like $\exp$ notation because of with this is easier to say that its inverse is $\ln$:
$(\exp \circ \ln)(x) = x$
A: The two notations are equivalent :


*

*the $exp(x)$ notation represents the exponential as a function,

*the $e^x$ notation shows the exponential as powers of Napier's constant.
There is no preference; the first one is better if you want to put a big expression as argument. Still if you use LaTeX you do not really care ...
