In the literature we find several different ways of introducing elementary functions. For the exponential function, we mention the following ways of characterising the exponential function:
(b) , also for complex values of x;
(c) x → exp (x) is the unique solution to the initial value problem 
(d) x → exp (x) is the inverse of
(e)x → exp (x) is the unique continuous function satisfying the
functional equation f (x + y) = f (x) f (y) and f(0) = 1 ; the corresponding definition is done for the logarithm in ;
(f) Define dr for rational r, and then use a continuity/density argument .
All of them have their advantages and disadvantages. We like (a) and (c), mostly because they have natural interpretations, (a) in the setting of compound interest and (c) being a simple model of many processes in physics and other sciences, but also because they are related to methods and ideas that are (usually) introduced rather early to the students.