The book "The Way of Analysis" by Robert Strichartz (Jones and Bartlett) has a nice discussion and comparison of these various was of defining exp and log functions. It's really a nice book.
In the book I am writing with Gabe Stolzenberg, we define the exponential function by extension of A^x on the rationals to A^x on the reals (i.e. by "completion"). Later we show it's equal to its power series using calculus. We approach the trig functions via power series (existence) and differential equations (properties). I also like this approach for exp and log and will probably make it an exercise in our book.