Editing Natural logarithm (section)

==Complex logarithms==
{{Main|Complex logarithm}}
The exponential function can be extended to a function which gives a [[complex number]] as {{math|''e''<sup>''z''</sup>}} for any arbitrary complex number {{mvar|z}}; simply use the infinite series with {{mvar|x}}=z complex. This exponential function can be inverted to form a complex logarithm that exhibits most of the properties of the ordinary logarithm. There are two difficulties involved: no {{mvar|x}} has {{math|1=''e''<sup>''x''</sup> = 0}}; and it turns out that {{math|1=''e''<sup>2''iπ''</sup> = 1 = ''e''<sup>0</sup>}}. Since the multiplicative property still works for the complex exponential function, {{math|1=''e''<sup>''z''</sup> = ''e''<sup>''z''+2''kiπ''</sup>}}, for all complex {{mvar|z}} and integers&nbsp;{{mvar|k}}.

So the logarithm cannot be defined for the whole [[complex plane]], and even then it is [[multi-valued]]—any complex logarithm can be changed into an "equivalent" logarithm by adding any integer multiple of {{math|2''iπ''}} at will. The complex logarithm can only be single-valued on the [[complex plane#Cutting the plane|cut plane]].  For example, {{math|ln ''i'' {{=}} {{sfrac|''iπ''|2}}}} or {{math|{{sfrac|5''iπ''|2}}}} or {{math|−{{sfrac|3''iπ''|2}}}}, etc.; and although {{math|''i''<sup>4</sup> {{=}} 1, 4 ln ''i''}} can be defined as {{math|2''iπ''}}, or {{math|10''iπ''}} or {{math|−6''iπ''}}, and so on.

<gallery mode="packed" caption="Plots of the natural logarithm function on the complex plane ([[principal branch]])">
Image:NaturalLogarithmRe.png|{{math|''z'' {{=}} Re(ln(''x'' + ''yi''))}} 
Image:NaturalLogarithmImAbs.png|{{math|''z'' {{=}} {{abs|(Im(ln(''x'' + ''yi'')))}}}} 
Image:NaturalLogarithmAbs.png|{{math|''z'' {{=}} {{abs|(ln(''x'' + ''yi''))}}}}
Image:NaturalLogarithmAll.png| Superposition of the previous three graphs
</gallery>