Are there reasons to prefer one definition of the exponential function over the other

calculusdefinitionexponential functionreal-analysis

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?

Best Answer

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:

  1. The unique differentiable function satisfying $\exp(0) = 1$ and $\exp'(x) = \exp(x)$.
  2. The inverse of the natural logarithm $\ln x = \int_1^x \frac{dt}{t}$.
  3. The function $\exp(x) = \lim_{n \to \infty} \left( 1 + \frac{x}{n} \right)^n$.
  4. 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 $e = \lim_{n \to \infty} \left( 1 + \frac{1}{n} \right)^n$.
  5. 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.

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