MATH215 Lecture Notes - Lecture 7: Rational Number

• School: University of Alberta
• Department: Mathematics
• Course: MATH215
• Professor: Alexander Litvak

Document Summary

An exponential function is a function of the form f (x) = ax, where a > 0. If x = n is a positive integer, then an = a a a n factors. If x = 0, then a0 = 1, and if x = n, where n is a positive integer, then a n = 1/an. If x = p/q is a rational number, where p and q are integers and q > 0, then ar = ap/q = q ap. In general, we de ne ax = lim r x ar, r rational. For a > 0, the exponential function ax is a continuous function with domain ( , ) and range (0, ). In particular, ax > 0 for all x. For a, b > 0 and x, y ( , ) we have ax ay = ax+y, ax ay = ax y, (ax)y = axy, and (ab)x = axbx.