Are there reasons to prefer one definition of the exponential function over the other? This question is motivated by curiosity and  I haven't much background to exhibit .
Going through a couple of books dealing with real analysis, I've noticed that 2 definitions can be given of the exponential function known in algebra as $f(x)= e^x$.
One definition says : The exponential function is the unique function defined on $\mathbb R$ such that $f(0)=1$ and $\forall (x) [   f'(x)= f(x)] $.
The other one defines the exponential function as the inverse of the natural logarithm function . More precisely $(1)$ $\exp_a (x)$ is defined as the inverse of $\log_a (x)$ , $(2)$ then , $\exp_a (x)$ is shown to be identical to $a^x$, and finally  $(3)$ every function of the form :  $a^x$  is shown to be a " special case" of the $e^x$ function.
My question :
(1) Do these definitions exhaust the ways the exponential function can be defined?
(2) Are these definitions actually different at least conceptually  ( though denoting in fact the same object)?
(3) Is there a reason to prefer one definition over the other? What is each definition good for?
 A: Title question:
Yes.

1. Do these definitions exhaust the ways the exponential function can be defined?
No. There are lots of other ways. For example:

*

*for any $a>0$ you can define $a^x$ as the unique continuous function $f_a:\Bbb{R}\to\Bbb{R}$ such that $f_a(1)=a$ and such that for all $x,y\in\Bbb{R}$, we have $f_a(x+y)=f_a(x)f_a(y)$. Uniqueness is relatively easy since such functions must agree on $\Bbb{Q}$, so by continuity, and density of $\Bbb{Q}$ in $\Bbb{R}$, must be equal on $\Bbb{R}$. See this answer for more details.

*You could just start with the series definition, $e^x=\sum_{n=1}^{\infty}\frac{x^n}{n!}$ (the way we arrive at this function is motivated by the IVP $f’=f, f(0)=1$, and by inserting a power-series ansatz for $f$), and show this series has infinite radius of convergence.

*you could try to show that for each $x\in\Bbb{R}$, the limit $\lim\limits_{n\to\infty}\left(1+\frac{x}{n}\right)^n$ exists, and you can call this $e^x$.

*you could first define for each $a>1$, the quantity $a^x$ when $x\in\Bbb{Q}$ (this isn’t too hard, though still requires analysis) and finally define for arbitrary $x\in\Bbb{R}$, $a^x:=\sup\{a^r\,:\, r\in\Bbb{Q}, r\leq x\}$. Finally, you can define the number $e:=\lim\limits_{n\to\infty}\left(1+\frac{1}{n}\right)^n$ (showing it first exists), and then define $e^x$ this way.


2. Are these definitions actually different at least conceptually ( though denoting in fact the same object)?
Afterall they’re the same thing, so answering whether they’re different conceptually is a subjective matter, but I’d say yes. For example, (1) is more of an “algebraic-analysis” question in the sense that it is asking for the non-trivial solution to a functional equation; this problem is “easy” on $\Bbb{Q}$, but more difficult on $\Bbb{R}$. Definition (4) is kind of similar, in that both (1) and (4) tell you essentially the same thing on the rationals, but then (4) is generalizing to the reals by preserving monotonicity.
Definition (3) is of course a separate definition, which can be motivated from the definition of the number $e$ given in definition (4), and some limit properties (which you need to prove).
Definition (2) is motivated more from an ODE standpoint.

3. Is there a reason to prefer one definition over the other? What is each definition good for?
Yes! I would prefer either the explicit series definition, or as the unique solution to the ODE. The reason to prefer one definition over another depends highly on the type of things you’re working on, and particularly so on the properties of the functions you’re looking to prove immediately.

*

*In an analysis class, the fact that $\exp’=\exp$ is of paramount importance, which is why you’ll often see it being explicitly defined using the power series, or by first defining $\log$ somehow (e.g. $\log(x)=\int_1^x\frac{1}{t}\,dt$ for $x>0$) and then inverting it (after showing it maps $(0,\infty)$ onto $\Bbb{R}$ bijectively). Next, after you learn about the existence and uniqueness theorems of ODEs, you can easily complete the proof of the equivalence of these three definitions. However, even here, I would argue that defining it via the series first, or using IVP is more fruitful because it gives an obvious generalization to complex analysis. Logarithms are a bit more iffy in the complex plane. Using any of these definitions, the fundamental property $\exp(x+y)=\exp(x)\exp(y)$ can easily be deduced, so it recovers definition (1) (after you define $e:=\exp(1)$). So, with these definitions, proving properties becomes very quick. Also, using the series or IVP allows for more generalization, even to the case of Banach algebras (which are extremely important and frequent in higher analysis). The only “downside” to this approach is that it requires the knowledge of power series (which you may consider advanced in terms of prerequisites, so unexplainable to elementary/middle-schoolers) and/or the theorems from ODE theory.

*Definition (1) takes a concept we know from middle/high school (in fact some kiddos “know” these “rules” so much so that they even apply those properties to numbers $a\in\Bbb{C}\setminus [0,\infty)$ and arrive at all sorts of “paradoxical” results), so in that sense it is a natural generalization based on the idea of “preserving the functional equation”. This definition has the advantage of keeping in-line with how exponentials are taught up to high school. The disadvantage is that to prove the existence of such a function, one needs to appeal to one of the other methods. Also, from here, proving differentiability is slightly annoying (also, defining $e$ is annoying here).

*Definition (4) is nice in the sense that if we accept that $a^x$ exists when $x$ is rational and $a>1$, and the monotonicity properties, then this definition still preserves monotonicity (so, it keeps our high-school dreams alive, hence it’s somewhat nice pedagogically). Also, once you learn about $\Bbb{R}$ as a complete ordered field, you’re ready to understand this definition (e.g Chapter 1 of Rudin’s Principles of Mathematical Analysis), so prerequisites wise, it is minimal. The disadvantage is that this is very tedious to work with. Proving the functional equation is a small mess, one still has to prove that $e$, given by the limit, is well-defined, proving continuity/differentiability properties are a pain etc. A more conceptual drawback of this definition is that it relies heavily on the total order of $\Bbb{R}$, hence doesn’t generalize to more general situations (e.g $\Bbb{C}$, or Banach-algebras more generally).

Also, back to my first bullet point, defining functions as solutions to ODEs may sound abstract, but it is actually very fruitful. Doing it for the exponential and trigonometric functions may sound silly because in middle/high-school we are introduced to them in a particular way, so the ODE approach seems out of the blue. However, once you do some analysis/ODEs you’ll find this very natural, because you’ll notice that very often certain ODEs just keep popping up, e.g the equation for a simple harmonic oscillator, or Bessel’s equation etc. We thus would like to investigate properties of their solutions (the trig functions, and Bessel functions respectively). Btw, here I show how we can derive the properties of trigonometric functions from the ODEs, and you’ll see that the treatment is pretty quick and also geometric (it’s slightly quicker to start with $e^z$ as a power series, and prove things directly from there).
Regarding definition (1) and my second bullet point here, functional equations are important in analysis, because not only does the exponential function satisfy such an important equation, but the Gamma function $\Gamma(z)$ (probably one of the more important special transcendental functions in analysis) also satisfies a very important functional equation, providing a smooth interpolation of the factorial (it’s not unique though… but it is unique if we require log-convexity; this is the Bohr-Mollerup theorem). See How can construction of the gamma function be motivated? for a discussion of this matter.

The bottom line is that it is good to have many ways of looking at the same thing, because it can give you further insight on your original way of thinking, and it can also give you different avenues for generalization. Of course you should try to understand each separately, and then see how they fit together.
A: There are several equivalent definitions, and it is important and valuable to know that they all define the same function. Really they should all be collected into a "definition-theorem," which might look like this.

Definition-Theorem: The following five functions are identical:



*

*The unique differentiable function satisfying $\exp(0) = 1$ and $\exp'(x) = \exp(x)$.

*The inverse of the natural logarithm $\ln x = \int_1^x \frac{dt}{t}$.

*The function $\exp(x) = \lim_{n \to \infty} \left( 1 + \frac{x}{n} \right)^n$.

*The function $e^x$ where $a^x$ is defined for rational $x$ and non-negative $a$ in the usual way and then extended by continuity to all real $x$, and where $e = \lim_{n \to \infty} \left( 1 + \frac{1}{n} \right)^n$.

*The function $\displaystyle \exp(x) = \sum_{n=0}^{\infty} \frac{x^n}{n!}$.


Of these, I personally favor introducing the exponential using definition 1. I think it offers the  most satisfying account of why $\exp(x)$ is a natural and interesting function to study: because it is an eigenvector of differentiation. This goes a long way towards explaining the role of $\exp(x)$ in solving differential equations which is where many of its applications are. However, it is a characterization rather than a construction: one has to do some additional work to show that such a function exists and is unique.
Definitions 3, 4, and 5 are all worth knowing but compared to definition 1 I think they lack motivation. In fact definition 1 is, again in my opinion, the best way to motivate them. Definition 3 arises from applying the Euler method to solve $f'(x) = f(x)$. Definition 5 arises from writing down a Taylor series solution to $f'(x) = f(x)$. And definition 4 provides no reason to single $e$ out over any other exponential base; that reason is provided by definition 1. On the other hand, these 3 definitions do successfully construct the exponential, which definition 1 does not without more background work.
Definition 2 is, I think, somewhat better motivated than 3, 4, or 5 but I still don't prefer it. The natural logarithm is a great and useful function but what motivates taking its inverse? The definition also requires having developed some theory of integration whereas definition 1 only requires derivatives. And, importantly, it does not readily generalize to the matrix exponential (also very useful for solving differential equations and a crucial tool in Lie theory), whereas definition 1 generalizes easily: for a fixed matrix $A$, the matrix exponential $t \mapsto \exp(tA)$ is the unique differentiable function $\mathbb{R} \to M_n(\mathbb{R})$ satisfying $\exp(0) = I$ and $\frac{d}{dt} \exp(tA) = A \exp(tA)$.
Definitions 3 and 5 also generalize to the matrix exponential whereas definition 4 does not. Even if you don't care about matrices yet this distinction is still relevant for complex exponentials, as in Euler's formula: neither definitions 2 nor 4 prepare you at all for understanding complex exponentials, whereas definitions 1, 3, and 5 generalize straightforwardly to this case as well.
However, definition 2 is notable for, I think, being closest to the historical line of development: natural logarithms were in fact discovered before either $e$ or the natural exponential. And definition 4 is notable in that it most directly connects the natural exponential to the ordinary pre-calculus exponential.
