Why did James Stewart regard it "more elegant" if you define the logarithm as an integral, and the exponential function its inverse? Stewart didn't explain why the "latter method" is "less intuitive", but I think this is because you can define the exponential function first, without integrals, that he does starting on p. 391. And integrals aren't intuitive. Correct?
Personally I like intuition. I just can't imagine how something can be "less intuitive but more elegant". If it's less intuitive, then it must less elegant!
I read What do we mean by an "Elegant Proof"?, but English is not my first language. Perhaps I don't understand the meaning of "elegant here".

There are two possible ways of treating the exponential and logarithmic functions and each
method has its passionate advocates. Because one often finds advocates of both approaches
teaching the same course, I include full treatments of both methods. In Sections 6.2, 6.3,
and 6.4 the exponential function is defined first, followed by the logarithmic function as
its inverse. (Students have seen these functions introduced this way since high school.) In
the alternative approach, presented in Sections 6.2*, 6.3*, and 6.4*, the logarithm is
defined as an integral and the exponential function is its inverse. This latter method is, of
course, less intuitive but more elegant. You can use whichever treatment you prefer.

James Stewart, Calculus 7th ed. 2011. Not the Early Transcendentals version. Page xiv from Preface.
 A: I'll give you a short overview of the process starting from both points:
$\ln$ first:
Define $\ln x:=\int_1^x\frac1t\mathrm dt$. Now the logarithm rules follow from the substitution rule of integrals (easy). Its differentiability and derivative follow from the fundamental theorem of calculus (reasonably easy). The logarithm is obviously monotonous, so it's injective. It's also surjective, since it's continuous by the fundamental theorem of calculus and unbounded above by comparison with the harmonic series and unbounded below by $\ln(1/x)=-\ln x$ (all reasonably easy), so its inverse $\exp$ exists on the reals. According to the inverse function rule, the inverse is differentiable (since $\ln$ is differentiable by the ftc) and has itself as its own derivative (easy). The rules for exponentiation follow from the logarithm rules (easy).
$\exp$ first:
First we must choose a definition. Shall it be the solution to the IVP $f'=f,~f(0)=1$? Then we must prove existence and uniqueness, which is guaranteed by Picard-Lindelöf (nontrivial theorem). But at least we have the rule for the derivative out of the way. Alternatively, define it as a power series. Now we have to prove that power series are differentiable, and to find the derivative we need to show that we can differentiate power series by differentiating each term individually (kinda non-trivial). Or we define it as $\exp x=\lim_{n\to\infty}\left(1+\frac xn\right)^n$. Now we also need to prove that it's differentiable, which not really hard but messy (you have to rearrange some not-so-nice sums with the binomial theorem). The rest is again reasonably easy (exponential rules, existence and differentiability of the inverse, etc.).
So essentially it boils down to the fact that proving the definitions of $\exp$ to yield sensible results is harder than to do the same for the definition of $\ln$.
A: Intuition and Elegance
The question seems to assert that the more intuitive an argument is, the more elegant it is.  I don't think that this is the way in which most mathematicians understand the terms.  Indeed, I think that "elegance" and "intuition" are often opposites of each other.  To explain:

*

*Intuition is about the way in which we understand how a theory is built.  An argument is intuitive if we can start from a solid foundation and then build up one step at a time, essentially "following our noses" through a line of reasoning.  Intuitive arguments are often easier to understand, but can get bogged down in computational detail.


*Elegance is about an economy of ideas.  A proxy for elegance is brevity:  in general, shorter arguments are considered more elegant.  I think that there is also a a kind of "Ah ha!" associated with elegance—elegant arguments often pull in seemingly unrelated arguments in a surprising way, and evoke some awe in the reader.
From this point of view, intuition and elegance are often at war with each other:  an intuitive idea rarely evokes awe, and often requires tedious computations in order to make rigorous, while elegant proofs often pull rabbits out of hats.
Of course, what is "intuitive" or "elegant" is very much a matter of opinion.  What is intuitive to one person may be utterly opaque and mysterious to another, and what I consider elegant might be a horrendous mess to another.
Working with Exponentials and Logarithms
Both logarithms and exponentials are important classes of functions, which were understood and studied separately at different moments in history.  Both classes of functions appear in relatively intuitive ways:  exponential functions show up as the solutions to certain "nice" differential equations, while logarithms provide a useful computational tool (the logarithm is a homomorphism from the multiplicative group of positive real numbers to the additive group of all real numbers—tables of logarithms were extremely useful before the advent of modern computers).
Regarding the introduction of logarithms and exponentials, I can see three basic approaches:

*

*Introduce exponential functions first, then define logarithms to be the inverses of exponentials.
Stewart calls this approach intuitive.  I am not sure that I agree, but I can, I think, make an argument in support of Stewart.  My recollection is that Stewart first defines exponential functions over the natural numbers: given $a > 0$ and $n \in \mathbb{N}$, define
$$
a^n = \prod_{j=1}^{n} a = \underbrace{a\cdot a\cdot \dotsb \cdot a}_{\text{$n$ times}}.
$$
That is, define exponentiation to be repeated multiplication.  This invokes very simple ideas which most American students learn in the second or third grade.
Using properties of this definition (e.g. $a^{m+n} = a^ma^n$, $(a^m)^n = a^{mn}$, and so on), it is possible to extend this definition to the rational numbers in a relatively straight-forward manner.  Then, because Stewart is writing a calculus text, he can extend this definition to the real numbers by continuity (it is necessary to show that such a continuous extension is possible, but this isn't too hard, and the underlying idea is central to calculus, so this is a reasonable next step).
Once this is done, we can note that the exponential function is strictly increasing, therefore one-to-one, therefore invertible.  Define the inverse to be the logarithm, and we're done.
The essential idea here is that we build up the theory in stages, one little bit at a time, using relatively intuitive ideas (properties of repeated multiplication, continuity, functional inverses, etc).  There is a "narrative arc" to the development of the theory, with each step following in a relatively intuitive way.  On the other hand, this approach may not be terribly elegant, as it requires a lot of little steps, each requiring some new mathematical tool.


*Introduce logarithmic functions first, then define exponentials to be the inverses of logarithms.
Defining the logarithm to be an antiderivative of $t \mapsto 1/t$ is not an "intuitive" or "obvious" thing to do (although I think that there is a narrative which can make it obvious, starting with John Napier, but this may be too much of a diversion from the material that a calculus instructor may want to introduce), but it is a computationally and theoretically "elegant" approach, in the sense that it doesn't require nearly the amount of boot-strapping as starting with the exponential.
In this approach, we define the function $\log : (0,\infty) \to \mathbb{R}$ by
$$
\log(x) := \int_{1}^{x} \frac{\mathrm{d}t}{t}.
$$
That this gives a well-defined function follows immediately from the fact that continuous functions are integrable over closed intervals (a theorem which should be old hat by the time we are working with logarithms), and most of the properties of a logarithm very quickly fall out of this definition.
Once the logarithm is defined, taking the exponential function to be the inverse (by definition) is a simple next step, and most of the properties of the exponential function follow fairly quickly from the properties of logarithms and/or the inverse function theorem.
There is an economy of ideas here which is, from a certain point of view, very elegant.  We really only need to know a couple of things (not much more than basic properties of the Riemann integral), but get a large number of important results and properties almost automatically.  This is not necessarily an obvious approach, but it requires very little development.


*Introduce exponential and logarithmic functions separately, then show that these functions are inverse to each other.
Personally, this is the approach that I prefer.  My feeling is that both exponential and logarithmic functions are natural objects to study, and so I like to study them separately, and then show that they are inverse to each other.  I typically introduce the exponential function via some variation on the approach mentioned above (essentially extend $\mathbb{Q} \to \mathbb{Q} : q \mapsto a^q$ by continuity to a function on $\mathbb{R}$, then consider the special value of $a$ such that $(a^x)' = a^x$).  This typically happens near the end of the discussion of derivatives—once we know a lot of things about differentiation, we can start talking about exponential functions.
Later, once the integral is fairly well understood, I introduce logarithms as described above.  I find it motivating to discuss this in the context of an "inverse power rule"—once the Fundamental Theorem of Calculus is proved, it is straightforward to show that if $q \ne -1$, then
$$ \int_{a}^{b} x^q \mathrm{d}x
= \frac{1}{q+1} x^{q+1}. $$
But this leaves a gap:  what happens when $q = -1$?  To investigate this, define a function $u : (0,\infty) \to \mathbb{R}$ by
$$ u(x) = \int_{1}^{x} t^{-1} \,\mathrm{d}t. $$
Again, we know that this defines a function, so explore the properties of this function.
Finally, having introduced the exponential and logarithm functions independently, it is not too difficult to show that they are inverse to each other.  I kind of like doing this, because it leads some students to say "Ah ha!" when the see the connection between these two, seemingly unrelated (families of) functions.
