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variants of this functions
Log




Mathematica Notation

Traditional Notation









Elementary Functions > Log[z] > Transformations > Products, sums, and powers of the direct function > Linear combinations of the direct function





http://functions.wolfram.com/01.04.16.0110.01









  


  








Input Form





a Log[x] + b Log[y] + c Log[z] == Log[x^a y^b z^c] - 2 I Pi Floor[(Pi - Im[a Log[x]] - Im[b Log[y]] - Im[c Log[z]])/(2 Pi)]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["a", " ", RowBox[List["Log", "[", "x", "]"]]]], "+", RowBox[List["b", " ", RowBox[List["Log", "[", "y", "]"]]]], "+", RowBox[List["c", " ", RowBox[List["Log", "[", "z", "]"]]]]]], "\[Equal]", RowBox[List[RowBox[List["Log", "[", RowBox[List[SuperscriptBox["x", "a"], " ", SuperscriptBox["y", "b"], " ", SuperscriptBox["z", "c"]]], "]"]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "-", RowBox[List["Im", "[", RowBox[List["a", " ", RowBox[List["Log", "[", "x", "]"]]]], "]"]], "-", RowBox[List["Im", "[", RowBox[List["b", " ", RowBox[List["Log", "[", "y", "]"]]]], "]"]], "-", RowBox[List["Im", "[", RowBox[List["c", " ", RowBox[List["Log", "[", "z", "]"]]]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> y </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> &#63449; </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msup> <mi> x </mi> <mi> a </mi> </msup> <mo> &#8290; </mo> <msup> <mi> y </mi> <mi> b </mi> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> c </mi> </msup> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mo> &#8970; </mo> <mfrac> <mrow> <mrow> <mo> - </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> x </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> y </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mi> &#960; </mi> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mfrac> <mo> &#8971; </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <ln /> <ci> x </ci> </apply> </apply> <apply> <times /> <ci> b </ci> <apply> <ln /> <ci> y </ci> </apply> </apply> <apply> <times /> <ci> c </ci> <apply> <ln /> <ci> z </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <ln /> <apply> <times /> <apply> <power /> <ci> x </ci> <ci> a </ci> </apply> <apply> <power /> <ci> y </ci> <ci> b </ci> </apply> <apply> <power /> <ci> z </ci> <ci> c </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <pi /> <apply> <floor /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <imaginary /> <apply> <times /> <ci> a </ci> <apply> <ln /> <ci> x </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <imaginary /> <apply> <times /> <ci> b </ci> <apply> <ln /> <ci> y </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <imaginary /> <apply> <times /> <ci> c </ci> <apply> <ln /> <ci> z </ci> </apply> </apply> </apply> </apply> <pi /> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["a_", " ", RowBox[List["Log", "[", "x_", "]"]]]], "+", RowBox[List["b_", " ", RowBox[List["Log", "[", "y_", "]"]]]], "+", RowBox[List["c_", " ", RowBox[List["Log", "[", "z_", "]"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["Log", "[", RowBox[List[SuperscriptBox["x", "a"], " ", SuperscriptBox["y", "b"], " ", SuperscriptBox["z", "c"]]], "]"]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "-", RowBox[List["Im", "[", RowBox[List["a", " ", RowBox[List["Log", "[", "x", "]"]]]], "]"]], "-", RowBox[List["Im", "[", RowBox[List["b", " ", RowBox[List["Log", "[", "y", "]"]]]], "]"]], "-", RowBox[List["Im", "[", RowBox[List["c", " ", RowBox[List["Log", "[", "z", "]"]]]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]]]]]]]


Date Added to functions.wolfram.com (modification date)





2007-05-02