Editing Exponentiation (section)

====Failure of power and logarithm identities====
Some identities for powers and logarithms for positive real numbers will fail for complex numbers, no matter how complex powers and complex logarithms are defined ''as single-valued functions''. For example:

{{bulleted list
| The identity {{math|1=log(''b''<sup>''x''</sup>) = ''x'' ⋅ log ''b''}} holds whenever {{mvar|b}} is a positive real number and {{mvar|x}} is a real number. But for the [[principal branch]] of the complex logarithm one has
<math display="block"> \log((-i)^2) = \log(-1) = i\pi \neq 2\log(-i) = 2\log(e^{-i\pi/2})=2\,\frac{-i\pi}{2} = -i\pi</math>

Regardless of which branch of the logarithm is used, a similar failure of the identity will exist. The best that can be said (if only using this result) is that:
<math display="block">\log w^z \equiv z \log w \pmod{2 \pi i}</math>

This identity does not hold even when considering log as a multivalued function. The possible values of {{math|log(''w''<sup>''z''</sup>)}} contain those of {{math|''z'' ⋅ log ''w''}} as a [[proper subset]]. Using {{math|Log(''w'')}} for the principal value of {{math|log(''w'')}} and {{mvar|m}}, {{mvar|n}} as any integers the possible values of both sides are:
<math display="block">\begin{align}
      \left\{\log w^z \right\} &= \left\{ z \cdot \operatorname{Log} w + z \cdot 2 \pi i n + 2 \pi i m \mid m,n\in\Z\right\} \\
      \left\{z \log w \right\} &= \left\{ z \operatorname{Log} w + z \cdot 2 \pi i n \mid n\in \Z\right\}
\end{align}</math>
| The identities {{math|1=(''bc'')<sup>''x''</sup> = ''b''<sup>''x''</sup>''c''<sup>''x''</sup>}} and {{math|1=(''b''/''c'')<sup>''x''</sup> = ''b''<sup>''x''</sup>/''c''<sup>''x''</sup>}} are valid when {{mvar|b}} and {{mvar|c}} are positive real numbers and {{mvar|x}} is a real number. But, for the principal values, one has
<math display="block">(-1 \cdot -1)^\frac{1}{2} =1 \neq (-1)^\frac{1}{2} (-1)^\frac{1}{2} =i \cdot i=i^2 =-1</math>
and
<math display="block">\left(\frac{1}{-1}\right)^\frac{1}{2} = (-1)^\frac{1}{2} = i \neq \frac{1^\frac{1}{2}}{(-1)^\frac{1}{2}} = \frac{1}{i} = -i</math>

On the other hand, when {{mvar|x}} is an integer, the identities are valid for all nonzero complex numbers.

If exponentiation is considered as a multivalued function then the possible values of {{math|(−1 ⋅ −1)<sup>1/2</sup>}} are {{math|{{mset|1, −1}}}}. The identity holds, but saying {{math|1={1} = {{mset|(−1 ⋅ −1)<sup>1/2</sup>}}}} is incorrect.
| The identity {{math|1=(''e''<sup>''x''</sup>)<sup>''y''</sup> = ''e''<sup>''xy''</sup>}} holds for real numbers {{mvar|x}} and {{mvar|y}}, but assuming its truth for complex numbers leads to the following [[mathematical fallacy|paradox]], discovered in 1827 by [[Thomas Clausen (mathematician)|Clausen]]:<ref name="Clausen1827">{{cite journal |author-last1=Steiner |author-first1=J. |author-last2=Clausen |author-first2=T. |author-last3=Abel |author-first3=Niels Henrik |author-link3=Niels Henrik Abel |title=Aufgaben und Lehrsätze, erstere aufzulösen, letztere zu beweisen |trans-title=Problems and propositions, the former to solve, the later to prove |journal=[[Crelle's Journal|Journal für die reine und angewandte Mathematik]] |volume=2 |date=1827 |pages=286–287 |url=https://www.digizeitschriften.de/dms/img/?PID=PPN243919689_0002%7Clog33&physid=phys301#navi}}</ref>

For any integer {{mvar|n}}, we have:
# <math>e^{1 + 2 \pi i n} = e^1 e^{2 \pi i n} = e \cdot 1 = e</math>
# <math>\left(e^{1 + 2\pi i n}\right)^{1 + 2 \pi i n} = e\qquad</math> (taking the <math>(1 + 2 \pi i n)</math>-th power of both sides)
# <math>e^{1 + 4 \pi i n - 4 \pi^2 n^2} = e\qquad</math> (using <math>\left(e^x\right)^y = e^{xy}</math> and expanding the exponent)
# <math>e^1 e^{4 \pi i n} e^{-4 \pi^2 n^2} = e\qquad</math> (using <math>e^{x+y} = e^x e^y</math>)
# <math>e^{-4 \pi^2 n^2} = 1\qquad</math> (dividing by {{mvar|e}})

but this is false when the integer {{mvar|n}} is nonzero.

The error is the following: by definition, <math>e^y</math> is a notation for <math>\exp(y),</math> a true function, and <math>x^y</math> is a notation for <math>\exp(y\log x),</math> which is a multi-valued function. Thus the notation is ambiguous when {{math|1=''x'' = ''e''}}. Here, before expanding the exponent, the second line should be
<math display="block">\exp\left((1 + 2\pi i n)\log \exp(1 + 2\pi i n)\right) = \exp(1 + 2\pi i n).</math>

Therefore, when expanding the exponent, one has implicitly supposed that <math>\log \exp z =z</math> for complex values of {{mvar|z}}, which is wrong, as the complex logarithm is multivalued. In other words, the wrong identity {{math|1=(''e''<sup>''x''</sup>)<sup>''y''</sup> = ''e''<sup>''xy''</sup>}} must be replaced by the identity
<math display="block">\left(e^x\right)^y = e^{y\log e^x},</math>
which is a true identity between multivalued functions.
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