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.
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.
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 ...