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 xay =  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 xay = 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.


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