Editing List of logarithmic identities (section)

=== Definitions ===

In what follows, a capital first letter is used for the principal value of functions, and the lower case version is used for the multivalued function. The single valued version of definitions and identities is always given first, followed by a separate section for the multiple valued versions.

*{{math|ln(''r'')}} is the standard [[natural logarithm]] of the real number {{mvar|r}}.
*{{math|Arg(''z'')}} is the principal value of the [[Arg (mathematics)|arg]] function; its value is restricted to {{open-closed|−''π'', ''π''}}. It can be computed using {{math|1=Arg(''x'' + ''iy'') = [[atan2]](''y'', ''x'')}}.
*{{math|Log(''z'')}} is the principal value of the complex logarithm function and has imaginary part in the range {{open-closed|−''π'', ''π''}}.
*<math>\operatorname{Log}(z) = \ln(|z|) + i \operatorname{Arg}(z)</math>
*<math>e^{\operatorname{Log}(z)} = z</math>

The multiple valued version of {{math|log(''z'')}} is a set, but it is easier to write it without braces and using it in formulas follows obvious rules.

*{{math|log(''z'')}} is the set of complex numbers ''v'' which satisfy {{math|1=e<sup>''v''</sup> = ''z''}}
*{{math|arg(''z'')}} is the set of possible values of the [[Arg (mathematics)|arg]] function applied to ''z''.

When ''k'' is any integer:

:<math>\log(z) = \ln(|z|) + i \arg(z)</math>
:<math>\log(z) = \operatorname{Log}(z) + 2 \pi i k</math>
:<math>e^{\log(z)} = z</math>