Log[z^a] == a Log[z] /; (z > 0 && Element[a, Reals]) || Inequality[-Pi, Less, Im[a Log[z]], LessEqual, Pi]

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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <msup> <mi> z </mi> <mi> a </mi> </msup> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> z </mi> <mo> &gt; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mi> a </mi> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[Reals]] </annotation> </semantics> </mrow> </mrow> <mo> &#8744; </mo> <mrow> <mrow> <mo> - </mo> <mi> &#960; </mi> </mrow> <mo> &lt; </mo> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8804; </mo> <mi> &#960; </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ln /> <apply> <power /> <ci> z </ci> <ci> a </ci> </apply> </apply> <apply> <times /> <ci> a </ci> <apply> <ln /> <ci> z </ci> </apply> </apply> </apply> <apply> <or /> <apply> <and /> <apply> <gt /> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <apply> <in /> <ci> a </ci> <reals /> </apply> </apply> <apply> <ci> Inequality </ci> <apply> <times /> <cn type='integer'> -1 </cn> <pi /> </apply> <lt /> <apply> <imaginary /> <apply> <times /> <ci> a </ci> <apply> <ln /> <ci> z </ci> </apply> </apply> </apply> <leq /> <pi /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Log", "[", SuperscriptBox["z_", "a_"], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["a", " ", RowBox[List["Log", "[", "z", "]"]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["z", ">", "0"]], "&&", RowBox[List["a", "\[Element]", "Reals"]]]], ")"]], "||", RowBox[List[RowBox[List["-", "\[Pi]"]], "<", RowBox[List["Im", "[", RowBox[List["a", " ", RowBox[List["Log", "[", "z", "]"]]]], "]"]], "\[LessEqual]", "\[Pi]"]]]]]]]]]]

2001-10-29