Editing Euler's formula (section)

==== Use of the formula to define the logarithm of complex numbers ====
Now, taking this derived formula, we can use Euler's formula to define the [[logarithm]] of a complex number. To do this, we also use the definition of the logarithm (as the inverse operator of exponentiation):
<math display="block">a = e^{\ln a}, </math>
and that
<math display="block">e^a e^b = e^{a + b}, </math>
both valid for any complex numbers {{mvar|a}} and {{mvar|b}}. Therefore, one can write:
<math display="block">z = \left|z\right| e^{i \varphi} = e^{\ln\left|z\right|} e^{i \varphi} = e^{\ln\left|z\right| + i \varphi}</math>
for any {{math|''z'' ≠ 0}}. Taking the logarithm of both sides shows that
<math display="block">\ln z = \ln \left|z\right| + i \varphi,</math>
and in fact, this can be used as the definition for the [[complex logarithm]]. The logarithm of a complex number is thus a [[multi-valued function]], because {{mvar|φ}} is multi-valued.

Finally, the other exponential law
<math display="block">\left(e^a\right)^k = e^{a k},</math>
which can be seen to hold for all integers {{mvar|k}}, together with Euler's formula, implies several [[trigonometric identities]], as well as [[de Moivre's formula]].