Log[(a + z)/(b + z)] == Log[(a + z)/(b - a)] - Log[(b + z)/(b - a)] /; Element[a - b, Reals] && a != b

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Log", "[", FractionBox[RowBox[List["a", "+", "z"]], RowBox[List["b", "+", "z"]]], "]"]], "\[Equal]", RowBox[List[RowBox[List["Log", "[", FractionBox[RowBox[List["a", "+", "z"]], RowBox[List["b", "-", "a"]]], "]"]], "-", RowBox[List["Log", "[", FractionBox[RowBox[List["b", "+", "z"]], RowBox[List["b", "-", "a"]]], "]"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["a", "-", "b"]], "\[Element]", "Reals"]], "\[And]", RowBox[List["a", "\[NotEqual]", "b"]]]]]]]]

<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> <mfrac> <mrow> <mi> a </mi> <mo> + </mo> <mi> z </mi> </mrow> <mrow> <mi> b </mi> <mo> + </mo> <mi> z </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mi> a </mi> <mo> + </mo> <mi> z </mi> </mrow> <mrow> <mi> b </mi> <mo> - </mo> <mi> a </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mi> b </mi> <mo> + </mo> <mi> z </mi> </mrow> <mrow> <mi> b </mi> <mo> - </mo> <mi> a </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> </mrow> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[Reals]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> a </mi> <mo> &#8800; </mo> <mi> b </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ln /> <apply> <times /> <apply> <plus /> <ci> a </ci> <ci> z </ci> </apply> <apply> <power /> <apply> <plus /> <ci> b </ci> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <ln /> <apply> <times /> <apply> <plus /> <ci> a </ci> <ci> z </ci> </apply> <apply> <power /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <apply> <times /> <apply> <plus /> <ci> b </ci> <ci> z </ci> </apply> <apply> <power /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <reals /> </apply> <apply> <neq /> <ci> a </ci> <ci> b </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Log", "[", FractionBox[RowBox[List["a_", "+", "z_"]], RowBox[List["b_", "+", "z_"]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["Log", "[", FractionBox[RowBox[List["a", "+", "z"]], RowBox[List["b", "-", "a"]]], "]"]], "-", RowBox[List["Log", "[", FractionBox[RowBox[List["b", "+", "z"]], RowBox[List["b", "-", "a"]]], "]"]]]], "/;", RowBox[List[RowBox[List[RowBox[List["a", "-", "b"]], "\[Element]", "Reals"]], "&&", RowBox[List["a", "\[NotEqual]", "b"]]]]]]]]]]

2001-10-29