Question

Why can the variable be raisesd only to a whole number? Why can't it be raised to any real no.s?

Answer 1

when the number you're raising to a power is positive. There are (at least) two ways to define a^x, where a>0 and x is a real number that may not be rational:

1. You can pick a sequence of rational numbers xnxn converging to x (i.e., limn→∞xn=x) and define

   ax=limn→∞axn.

2. You can use the exponential function ex (defined in many ways, say as ex=limn→∞(1+x/n)n or with a power series), and its inverse the logarithmic function that satifies elnt=t for all positive t, and since a=e^lna, define

   a^x=e^xlna.