Log[a, c z] == Log[a, c] + Log[a, z] /; c > 0

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Log", "[", RowBox[List["a", ",", RowBox[List["c", " ", "z"]]]], "]"]], "\[Equal]", RowBox[List[RowBox[List["Log", "[", RowBox[List["a", ",", "c"]], "]"]], "+", RowBox[List["Log", "[", RowBox[List["a", ",", "z"]], "]"]]]]]], "/;", " ", RowBox[List["c", ">", "0"]]]]]]

<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> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <msub> <mi> log </mi> <mi> a </mi> </msub> <mo> ( </mo> <mi> c </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <msub> <mi> log </mi> <mi> a </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> c </mi> <mo> &gt; </mo> <mn> 0 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <log /> <logbase> <ci> a </ci> </logbase> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <log /> <logbase> <ci> a </ci> </logbase> <ci> c </ci> </apply> <apply> <log /> <logbase> <ci> a </ci> </logbase> <ci> z </ci> </apply> </apply> </apply> <apply> <gt /> <ci> c </ci> <cn type='integer'> 0 </cn> </apply> </apply> </annotation-xml> </semantics> </math>

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

2001-10-29