Log[a, z^c] == c Log[a, z] /; z > 0 && Element[c, Reals]

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Log", "[", RowBox[List["a", ",", SuperscriptBox["z", "c"]]], "]"]], "\[Equal]", RowBox[List["c", " ", RowBox[List["Log", "[", RowBox[List["a", ",", "z"]], "]"]]]]]], "/;", RowBox[List[RowBox[List["z", ">", "0"]], " ", "\[And]", RowBox[List["c", "\[Element]", "Reals"]]]]]]]]

<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> z </mi> <mi> c </mi> </msup> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mrow> <msub> <mi> log </mi> <mi> a </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> z </mi> <mo> &gt; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mi> c </mi> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[Reals]] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <log /> <logbase> <ci> a </ci> </logbase> <apply> <power /> <ci> z </ci> <ci> c </ci> </apply> </apply> <apply> <times /> <ci> c </ci> <apply> <log /> <logbase> <ci> a </ci> </logbase> <ci> z </ci> </apply> </apply> </apply> <apply> <and /> <apply> <gt /> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <apply> <in /> <ci> c </ci> <reals /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Log", "[", RowBox[List["a_", ",", SuperscriptBox["z_", "c_"]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["c", " ", RowBox[List["Log", "[", RowBox[List["a", ",", "z"]], "]"]]]], "/;", RowBox[List[RowBox[List["z", ">", "0"]], "&&", RowBox[List["c", "\[Element]", "Reals"]]]]]]]]]]

2001-10-29