Why is Euler's number used as a base for logarithms?
Base 10 would be arbitrary, though it is more helpful in getting a feeling of the order of magnitude of the original numbers. The reason to pick base $e$ is that one can define $\ln$ naturally even without picking a base, for example as $$\ln x:=\int_1^x\frac{\mathrm dt}t$$ Another (or maybe the same?) advantage is that it is easy to estimate $\ln x$ for numbers close to $1$, for we have $\ln(1+h)\approx h$ if $h$ is small. Even the complete Taylor series for $\ln(1+x)$ is nice: $\ln(1+x)=x-\frac12x^2+\frac13x^3-\frac14x^4\pm\ldots$, in particular, all the coefficients are nice and rational. When working with any other base you'd have to carry an irrational constant around all the time, whereas with the natural log, this irrational constant is visible only when you write $\log_e$ instead of $\ln$.
It was suggested to consider this question as a duplicate of What's so “natural” about the base of natural logarithms?. But frankly, looking at the accepted answer:
If you know some linear algebra, then here is an abstract reason: $e^x$ is the unique eigenvector of eigenvalue $1$ of the derivative $D$ acting on, say, the space of smooth functions on $\mathbb{R}$.
one could argue that someone who understands this answer would likely not have asked the question in the first place. ("Eigen...what?")
For me (as a non-mathematician), the key property of this number is, geometrically speaking, that the graph of $e^x$ is describing its own slope. I tried to point this out in these images:
A little bit more formally, this means that $e$ is the only number that has the property
$$ \frac{d}{dx} e^x = e^x$$
which can be seen when plotting both functions.
Certain higher-level mathematical statements (e.g. that $e^x$ is this eigen...vector) are consequences of that.
Historically, I believe it was because it was much easier to compute.
How do you compute $10^{4.32}$? Maybe you can raise $10^{432}$, but then how do you extract that $100^{th}$ root? Maybe you use Newton's method or something, but this is a long process, and it is not "uniform": making a table of values this way would be horrible.
However, to approximate a table of values for $e^x$ is pretty easy:
$\begin{align*} e^{0.1} &\approx \left((1+\frac{1}{10})^{10} \right)^{0.1}=1.1\\ e^{0.2} &\approx 1.1^2 = 1.21\\ e^{0.3} &\approx 1.1^3 = 1.331\\ \vdots \end{align*}$
You can improve the accuracy systematically by using finer approximations, like $e^{0.01} \approx 1.01$. All you have to do to generate the table is multiply (which for these bases amounts to shifting and adding).
Napier used $e^{0.00000001}$ to make his table of logarithms. Handling the error in the terms (which get small fast, but add up over the course of a very long table) was highly nontrivial. Once he has his log tables, it allowed a huge number of computations to be done more easily (including the creation of tools like the slide rule).