For x = m/n and a > 0,  a ^x = a ^m/n = exp( (m/n) ln(a)) = exp( x ln(a)). This suggests putting a ^x = exp( x ln(a)) for x irrational.  Then

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

and not only for rational numbers. From this definition,  ln a ^x =  x ln(a).  Therefore log_a(a ^x) = x  because  log_a(x) = ln(x)/ln(a).

Related posts:

1. Natural Logarithms and Exponentials – Roots and Powers