Now (a ^x)^y = exp(y ln(a ^x )) =   exp(y x ln(a )) =  a ^yx = a ^xy Therefore

(a ^x)^y =  a ^xy (Exponential of an exponential)

Now a ^xa^y =  exp(x ln(a)) · exp(y ln(a) = exp(x ln(a)+y ln(a)) = exp( (x +y )ln(a) ) = a ^x+y Therefore  has the exponential property

a ^xa^y = a ^x+y for all real numbers x and y when a > 0.

Now  for the natural number e = exp(1) = 2.718281828…  (irrational, deci), the natural logarithm of e,  ln (e) = 1 Therefore

e ^x = exp( x) for all real x when a > 0

as a ^x = exp( x ln(a)). Calculators often have a button marked e ^x for the evaluation of the exponential function exp( x)

Caution: the capital EXP on some calculators will not help you with the calculation of exp(x). Use the  button marked e ^x instead.

Related posts:

1. Natural Logarithms and Exponentials – Roots and Powers
2. Exponentials of Real Numbers a x = exp( x ln(a))