section_path: Real exponents/Powers via logarithms
section_name: Powers via logarithms
paragraph_text: Powers via logarithms
  
  The definition of e as the exponential function allows defining b for every positive real numbers b, in terms of exponential and logarithm function. Specifically, the fact that the natural logarithm ln(x) is the inverse of the exponential function e means that one has
  
  for every b > 0. For preserving the identity (e x) y = e x y, {\displaystyle (e^{x})^{y}=e^{xy},} one must have
  
  So, e x ln b {\displaystyle e^{x\ln b}} can be used as an alternative definition of b for any positive real b. This agrees with the definition given above using rational exponents and continuity, with the advantage to extend straightforwardly to any complex exponent.