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

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Log", "[", RowBox[List["a", ",", SuperscriptBox["a", "z"]]], "]"]], "\[Equal]", "z"]], "/;", RowBox[List[RowBox[List[RowBox[List["a", "\[Element]", "Reals"]], "\[And]", RowBox[List["z", ">", "0"]]]], "\[Or]", RowBox[List["(", RowBox[List[RowBox[List["a", ">", "0"]], "\[And]", RowBox[List[RowBox[List["-", "\[Pi]"]], "<", RowBox[List["Im", "[", RowBox[List["z", " ", RowBox[List["Log", "[", "a", "]"]]]], "]"]], "\[LessEqual]", "\[Pi]"]]]], ")"]]]]]]]]

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

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

2001-10-29