text: If  is real and positive, the principal value of  equals its usual value defined above. If  where  is an integer, this principal value is the same as the one defined above.
  Multivalued function
  In some contexts, there is a problem with the discontinuity of the principal values of  and  at the negative real values of . In this case, it is useful to consider these functions as multivalued functions.
  If  denotes one of the values of the multivalued logarithm (typically its principal value), the other values are  where  is any integer. Similarly, if  is one value of the exponentiation, then the other values are given by
  where  is any integer.
  Different values of  give different values of  unless  is a rational number, that is, there is an integer  such that  is an integer. This results from the periodicity of the exponential function, more specifically, that  if and only if  is an integer multiple of
  If  is a rational number with  and  coprime integers with  then  has exactly  values. In the case  these values are the same as those described in § th roots of a complex number. If  is an integer, there is only one value that agrees with that of .