Do you know of any Calculus text defining the exponential and logarithm functions in an alternative way? The story in nearly every introductory Calculus book is well known by everybody: you don't have the "right" to raise a number to an irrational power, so forget exponents for now and let's take a look at $y=x^{-1}$. How odd, the innocent formula for a power function's antiderivative breaks down but gee, it must have an antiderivative, it's smooth! Let's examine its properties...
...and in the end, Rosebud was his sled, no, wait, the mysterious antiderivative turns out to have an inverse that corresponds exactly to the elementary school concept of exponents, only it works for irrational exponents too! The hero wins! The End.
But what if we start from the opposite end? Start with the innocent, only-defined-for-rationals (so far) exponential function $y=k^x$, $k>0$, and if $x_0$ is irrational, prove that, as $x$ (while staying rational) approaches $x_0$, $k^x$ approaches some specific real number. Define that such number is $k^{x_0}$.
And from there, prove that our New! Improved! $k^x$ is continuous, has a derivative that's also an exponential, that there is some $k=e$ for which the exponential is its own derivative, that the inverse of $e^x$ has $x^{-1}$ as its derivative etc etc...
Do you know of any Calculus text that takes that approach?

I like the approach Lang takes in Undergraduate Analysis. He defines the exponential as a function that satisfies the following differential equation subject to specified initial conditions:
$$ 

f^{\prime} = f, \;f(0)=1

$$
Using these assumptions he shows that if $f$ exists then it is unique. Later in the text he proves existence with power series. He gives an analagous treatment for $sin(x)$ and $cos(x)$. Fitzpatrick's Advanced Calculus takes a similar approach 
