(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.2' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 64089, 1939]*) (*NotebookOutlinePosition[ 64799, 1964]*) (* CellTagsIndexPosition[ 64755, 1960]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Exercices - Logarithmes et exponentielles", "Section"], Cell[BoxData[{\(Unprotect[Log];\), "\[IndentingNewLine]", \(Log /: Format[Log[x_]] := \ ln[x];\), "\[IndentingNewLine]", \(EnSol[x \[LessEqual] a_] := StringForm["\<\[LongLeftArrow] , ``]\>", a];\), "\[IndentingNewLine]", RowBox[{\(EnSol[x < a_] := StringForm["\<\[LongLeftArrow] , `` [\>", a];\), "\[IndentingNewLine]"}], "\[IndentingNewLine]", 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InterpretationBox[ RowBox[{ "14", "\[InvisibleSpace]", "\<\") \"\>", "\[InvisibleSpace]", RowBox[{ FormBox[\(ln(x\^2 + 5\ x + 6)\), "TraditionalForm"], "\[LongEqual]", FormBox[\(ln(x + 11)\), "TraditionalForm"]}]}], SequenceForm[ 14, ") ", Equal[ Log[ Plus[ 6, Times[ 5, x], Power[ x, 2]]], Log[ Plus[ 11, x]]]], Editable->False], TraditionalForm]], "Print"], Cell[BoxData[ FormBox[ InterpretationBox[ RowBox[{ "15", "\[InvisibleSpace]", "\<\") \"\>", "\[InvisibleSpace]", RowBox[{ RowBox[{ TagBox[\(\(log\_3\)(4\ x + 1)\), HoldForm], "+", TagBox[\(\(log\_3\)(x + 2)\), HoldForm], "-", RowBox[{"2", " ", TagBox[\(\(log\_3\)(3\ x)\), HoldForm]}]}], "\[LongEqual]", "0"}]}], SequenceForm[ 15, ") ", Equal[ Plus[ HoldForm[ Log[ 3, Plus[ Times[ 4, x], 1]]], HoldForm[ Log[ 3, Plus[ x, 2]]], Times[ -2, HoldForm[ Log[ 3, Times[ 3, x]]]]], 0]], Editable->False], TraditionalForm]], "Print"], Cell[BoxData[ FormBox[ InterpretationBox[ RowBox[{ "16", "\[InvisibleSpace]", "\<\") \"\>", "\[InvisibleSpace]", RowBox[{ RowBox[{ RowBox[{"2", " ", FormBox[\(ln(1 - x)\), "TraditionalForm"]}], "-", FormBox[\(ln(x + 1)\), "TraditionalForm"], "+", FormBox[\(ln(2\ x + 3)\), "TraditionalForm"]}], "\[LongEqual]", RowBox[{"2", " ", FormBox[\(ln(3)\), "TraditionalForm"]}]}]}], SequenceForm[ 16, ") ", Equal[ Plus[ Times[ 2, Log[ Plus[ 1, Times[ -1, x]]]], Times[ -1, Log[ Plus[ 1, x]]], Log[ Plus[ 3, Times[ 2, x]]]], Times[ 2, Log[ 3]]]], Editable->False], TraditionalForm]], "Print"], Cell[BoxData[ FormBox[ InterpretationBox[ RowBox[{ "17", "\[InvisibleSpace]", "\<\") \"\>", "\[InvisibleSpace]", RowBox[{ RowBox[{ TagBox[\(\(log\_10\)(x\^2 + 2\ x - 3)\), HoldForm], "-", RowBox[{"2", " ", TagBox[\(\(log\_10\)(x - 1)\), HoldForm]}]}], "\[LongEqual]", "2"}]}], SequenceForm[ 17, ") ", Equal[ Plus[ HoldForm[ Log[ 10, Plus[ Power[ x, 2], Times[ 2, x], -3]]], Times[ -2, HoldForm[ Log[ 10, Plus[ x, -1]]]]], 2]], Editable->False], TraditionalForm]], "Print"], Cell[BoxData[ FormBox[ InterpretationBox[ RowBox[{ "18", "\[InvisibleSpace]", "\<\") \"\>", "\[InvisibleSpace]", RowBox[{ RowBox[{ FormBox[\(ln(x - 1)\), "TraditionalForm"], "+", FormBox[\(ln(x)\), "TraditionalForm"]}], "\[LongEqual]", FormBox[\(ln(x + 3)\), "TraditionalForm"]}]}], SequenceForm[ 18, ") ", Equal[ Plus[ Log[ Plus[ -1, x]], Log[ x]], Log[ Plus[ 3, x]]]], Editable->False], TraditionalForm]], "Print"], Cell[BoxData[ FormBox[ InterpretationBox[ RowBox[{ "19", "\[InvisibleSpace]", "\<\") \"\>", "\[InvisibleSpace]", RowBox[{ RowBox[{ RowBox[{"3", " ", FormBox[\(ln(x)\), "TraditionalForm"]}], "-", FormBox[\(ln(x + 2)\), "TraditionalForm"]}], "\[LongEqual]", FormBox[\(ln(2)\), "TraditionalForm"]}]}], SequenceForm[ 19, ") ", Equal[ Plus[ Times[ 3, Log[ x]], Times[ -1, Log[ Plus[ 2, x]]]], Log[ 2]]], Editable->False], TraditionalForm]], "Print"], Cell[BoxData[ FormBox[ InterpretationBox[\(20\[InvisibleSpace]") "\[InvisibleSpace]\(15 - 11\ \[ExponentialE]\^x + 2\ \[ExponentialE]\^\(2\ x\) \[LongEqual] 0\)\), SequenceForm[ 20, ") ", Equal[ Plus[ 15, Times[ -11, Power[ E, x]], Times[ 2, Power[ E, Times[ 2, x]]]], 0]], Editable->False], TraditionalForm]], "Print"], Cell[BoxData[ FormBox[ InterpretationBox[\(21\[InvisibleSpace]") "\[InvisibleSpace]\(6 + 4\^x \[LongEqual] 5\ 2\^x\)\), SequenceForm[ 21, ") ", Equal[ Plus[ 6, Power[ 4, x]], Times[ 5, Power[ 2, x]]]], Editable->False], TraditionalForm]], "Print"], Cell[BoxData[ FormBox[ InterpretationBox[ RowBox[{ "22", "\[InvisibleSpace]", "\<\") \"\>", "\[InvisibleSpace]", RowBox[{ RowBox[{ FormBox[\(ln(x)\), "TraditionalForm"], " ", RowBox[{"(", RowBox[{ FormBox[\(ln(x)\), "TraditionalForm"], "+", "2"}], ")"}]}], "\[LongEqual]", \(-1\)}]}], SequenceForm[ 22, ") ", Equal[ Times[ Log[ x], Plus[ 2, Log[ x]]], -1]], Editable->False], TraditionalForm]], "Print"], Cell[BoxData[ FormBox[ InterpretationBox[ RowBox[{ "23", "\[InvisibleSpace]", "\<\") \"\>", "\[InvisibleSpace]", RowBox[{ RowBox[{ FormBox[\(ln(x)\), "TraditionalForm"], " ", RowBox[{"(", RowBox[{ RowBox[{"3", " 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x]]]], 1]], Editable->False], TraditionalForm]], "Print"], Cell[BoxData[ FormBox[ InterpretationBox[\(27\[InvisibleSpace]") "\[InvisibleSpace]\(\(-\(1\/\ \[ExponentialE]\)\) + \[ExponentialE]\^x \[GreaterEqual] 0\)\), SequenceForm[ 27, ") ", GreaterEqual[ Plus[ Times[ -1, Power[ E, -1]], Power[ E, x]], 0]], Editable->False], TraditionalForm]], "Print"], Cell[BoxData[ FormBox[ InterpretationBox[ RowBox[{ "28", "\[InvisibleSpace]", "\<\") \"\>", "\[InvisibleSpace]", RowBox[{ TagBox[\(\(log\_\(1\/2\)\)(x + 4)\), HoldForm], ">", TagBox[\(\(log\_\(1\/2\)\)(2\ x - 3)\), HoldForm]}]}], SequenceForm[ 28, ") ", Greater[ HoldForm[ Log[ Times[ 1, Power[ 2, -1]], Plus[ x, 4]]], HoldForm[ Log[ Times[ 1, Power[ 2, -1]], Plus[ Times[ 2, x], -3]]]]], Editable->False], TraditionalForm]], "Print"], Cell[BoxData[ FormBox[ InterpretationBox[ RowBox[{ "29", "\[InvisibleSpace]", "\<\") \"\>", "\[InvisibleSpace]", RowBox[{ FormBox[\(ln(x\^2 + 3)\), "TraditionalForm"], "\[GreaterEqual]", RowBox[{ FormBox[\(ln(x)\), "TraditionalForm"], "+", RowBox[{"2", " ", FormBox[\(ln(2)\), "TraditionalForm"]}]}]}]}], SequenceForm[ 29, ") ", GreaterEqual[ Log[ Plus[ 3, Power[ x, 2]]], Plus[ Times[ 2, Log[ 2]], Log[ x]]]], Editable->False], TraditionalForm]], "Print"], Cell[BoxData[ FormBox[ InterpretationBox[\(30\[InvisibleSpace]") "\[InvisibleSpace]\(49\^x \ \[GreaterEqual] 7\^\(x - 1\)\)\), SequenceForm[ 30, ") ", GreaterEqual[ Power[ 49, x], Power[ 7, Plus[ -1, x]]]], Editable->False], TraditionalForm]], "Print"], Cell[BoxData[ FormBox[ InterpretationBox[\(31\[InvisibleSpace]") "\[InvisibleSpace]\(\(-9\) + \ \[ExponentialE]\^\(3\ x - 1\) > 0\)\), SequenceForm[ 31, ") ", Greater[ Plus[ -9, Power[ E, Plus[ -1, Times[ 3, x]]]], 0]], Editable->False], TraditionalForm]], "Print"], Cell[BoxData[ FormBox[ InterpretationBox[\(32\[InvisibleSpace]") "\[InvisibleSpace]\(4 - 5\ 2\^x + 2\^\(2\ x\) \[LessEqual] 0\)\), SequenceForm[ 32, ") ", LessEqual[ Plus[ 4, Times[ -5, Power[ 2, x]], Power[ 2, Times[ 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FormBox[\(ln(x)\), "TraditionalForm"], "+", "1"}], RowBox[{ FormBox[\(ln(x)\), "TraditionalForm"], "-", "1"}]], "<", "0"}]}], SequenceForm[ 35, ") ", Less[ Times[ Power[ Plus[ -1, Log[ x]], -1], Plus[ 1, Log[ x]]], 0]], Editable->False], TraditionalForm]], "Print"] }, Open ]], Cell[CellGroupData[{ Cell["Solutions", "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ \(\(Do[Print[i, "\<) S = \>", solution[i]], {i, 35}];\)\)], "Input", CellOpen->False], Cell[BoxData[ FormBox[ InterpretationBox[ RowBox[{ "1", "\[InvisibleSpace]", "\<\") S = \"\>", "\[InvisibleSpace]", InterpretationBox["\<\"{\\!\\(TraditionalForm\\`0\\)}\"\>", StringForm[ "{``}", 0], Editable->False]}], SequenceForm[ 1, ") S = ", StringForm[ "{``}", 0]], Editable->False], TraditionalForm]], "Print"], Cell[BoxData[ FormBox[ InterpretationBox[ RowBox[{ "2", "\[InvisibleSpace]", "\<\") S = \"\>", "\[InvisibleSpace]", InterpretationBox["\<\"{\\!\\(TraditionalForm\\`\\(TraditionalForm\\\ `\\(ln(4)\\)\\)\\)}\"\>", StringForm[ 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"\[InvisibleSpace]", "\<\") S = \"\>", "\[InvisibleSpace]", InterpretationBox["\<\"{\\!\\(TraditionalForm\\`2\\)}\"\>", StringForm[ "{``}", 2], Editable->False]}], SequenceForm[ 8, ") S = ", StringForm[ "{``}", 2]], Editable->False], TraditionalForm]], "Print"], Cell[BoxData[ FormBox[ InterpretationBox[ RowBox[{ "9", "\[InvisibleSpace]", "\<\") S = \"\>", "\[InvisibleSpace]", InterpretationBox["\<\"{\\!\\(TraditionalForm\\`3\\)}\"\>", StringForm[ "{``}", 3], Editable->False]}], SequenceForm[ 9, ") S = ", StringForm[ "{``}", 3]], Editable->False], TraditionalForm]], "Print"], Cell[BoxData[ FormBox[ InterpretationBox[ RowBox[{ "10", "\[InvisibleSpace]", "\<\") S = \"\>", "\[InvisibleSpace]", InterpretationBox["\<\"{\\!\\(TraditionalForm\\`9\\)}\"\>", StringForm[ "{``}", 9], Editable->False]}], SequenceForm[ 10, ") S = ", StringForm[ "{``}", 9]], Editable->False], TraditionalForm]], "Print"], Cell[BoxData[ FormBox[ InterpretationBox[ RowBox[{ "11", "\[InvisibleSpace]", "\<\") S = \"\>", "\[InvisibleSpace]", InterpretationBox["\<\"{\\!\\(TraditionalForm\\`0\\)}\"\>", StringForm[ "{``}", 0], Editable->False]}], SequenceForm[ 11, ") S = ", StringForm[ "{``}", 0]], Editable->False], TraditionalForm]], "Print"], Cell[BoxData[ FormBox[ InterpretationBox[ RowBox[{ "12", "\[InvisibleSpace]", "\<\") S = \"\>", "\[InvisibleSpace]", InterpretationBox["\<\"{\\!\\(TraditionalForm\\`7\\)}\"\>", StringForm[ "{``}", 7], Editable->False]}], SequenceForm[ 12, ") S = ", StringForm[ "{``}", 7]], Editable->False], TraditionalForm]], "Print"], Cell[BoxData[ FormBox[ InterpretationBox[ RowBox[{ "13", "\[InvisibleSpace]", "\<\") S = \"\>", "\[InvisibleSpace]", InterpretationBox["\<\"{\\!\\(TraditionalForm\\`\\(1\\/8\\\\ \ \\(\\((33 + \\@65)\\)\\)\\)\\)}\"\>", StringForm[ "{``}", Times[ Rational[ 1, 8], Plus[ 33, Power[ 65, Rational[ 1, 2]]]]], Editable->False]}], SequenceForm[ 13, ") S = ", StringForm[ "{``}", Times[ Rational[ 1, 8], Plus[ 33, Power[ 65, Rational[ 1, 2]]]]]], Editable->False], TraditionalForm]], "Print"], Cell[BoxData[ FormBox[ InterpretationBox[ RowBox[{ "14", "\[InvisibleSpace]", "\<\") S = \"\>", "\[InvisibleSpace]", InterpretationBox["\<\"{\\!\\(TraditionalForm\\`\\(-5\\)\\) , \ \\!\\(TraditionalForm\\`1\\)}\"\>", StringForm[ "{`1` , `2`}", -5, 1], Editable->False]}], SequenceForm[ 14, ") S = ", StringForm[ "{`1` , `2`}", -5, 1]], Editable->False], TraditionalForm]], "Print"], Cell[BoxData[ FormBox[ InterpretationBox[ RowBox[{ "15", "\[InvisibleSpace]", "\<\") S = \"\>", "\[InvisibleSpace]", InterpretationBox["\<\"{\\!\\(TraditionalForm\\`2\\)}\"\>", StringForm[ "{``}", 2], Editable->False]}], SequenceForm[ 15, ") S = ", StringForm[ "{``}", 2]], Editable->False], TraditionalForm]], "Print"], Cell[BoxData[ FormBox[ InterpretationBox[ RowBox[{ "16", "\[InvisibleSpace]", "\<\") S = \"\>", "\[InvisibleSpace]", InterpretationBox["\<\"{\\!\\(TraditionalForm\\`\\(-\\(\\(1\\/2\\)\\\ )\\)\\)}\"\>", StringForm[ "{``}", Rational[ -1, 2]], Editable->False]}], SequenceForm[ 16, ") S = ", StringForm[ "{``}", Rational[ -1, 2]]], Editable->False], TraditionalForm]], "Print"], Cell[BoxData[ FormBox[ InterpretationBox[ RowBox[{ "17", "\[InvisibleSpace]", "\<\") S = \"\>", "\[InvisibleSpace]", InterpretationBox["\<\"{\\!\\(TraditionalForm\\`103\\/99\\)}\"\>", StringForm[ "{``}", Rational[ 103, 99]], Editable->False]}], SequenceForm[ 17, ") S = ", StringForm[ "{``}", Rational[ 103, 99]]], Editable->False], TraditionalForm]], "Print"], Cell[BoxData[ FormBox[ InterpretationBox[ RowBox[{ "18", "\[InvisibleSpace]", "\<\") S = \"\>", "\[InvisibleSpace]", InterpretationBox["\<\"{\\!\\(TraditionalForm\\`3\\)}\"\>", StringForm[ "{``}", 3], Editable->False]}], SequenceForm[ 18, ") S = ", StringForm[ "{``}", 3]], Editable->False], TraditionalForm]], "Print"], Cell[BoxData[ FormBox[ InterpretationBox[ RowBox[{ "19", "\[InvisibleSpace]", "\<\") S = \"\>", "\[InvisibleSpace]", InterpretationBox["\<\"{\\!\\(TraditionalForm\\`2\\)}\"\>", StringForm[ "{``}", 2], Editable->False]}], SequenceForm[ 19, ") S = ", StringForm[ "{``}", 2]], Editable->False], TraditionalForm]], "Print"], Cell[BoxData[ FormBox[ InterpretationBox[ RowBox[{ "20", "\[InvisibleSpace]", "\<\") S = \"\>", "\[InvisibleSpace]", InterpretationBox["\<\"{\\!\\(TraditionalForm\\`\\(TraditionalForm\\\ `\\(ln(\\(\\(5\\/2\\)\\))\\)\\)\\) , \ \\!\\(TraditionalForm\\`\\(TraditionalForm\\`\\(ln(3)\\)\\)\\)}\"\>", StringForm[ "{`1` , `2`}", Log[ Rational[ 5, 2]], Log[ 3]], Editable->False]}], SequenceForm[ 20, ") S = ", StringForm[ "{`1` , `2`}", Log[ Rational[ 5, 2]], Log[ 3]]], Editable->False], TraditionalForm]], "Print"], Cell[BoxData[ FormBox[ InterpretationBox[ RowBox[{ "21", "\[InvisibleSpace]", "\<\") S = \"\>", "\[InvisibleSpace]", InterpretationBox["\<\"{\\!\\(TraditionalForm\\`1\\) , \ \\!\\(TraditionalForm\\`\\(TraditionalForm\\`\\(ln(3)\\)\\)\\/\\(\ TraditionalForm\\`\\(ln(2)\\)\\)\\)}\"\>", StringForm[ "{`1` , `2`}", 1, Times[ Power[ Log[ 2], -1], Log[ 3]]], Editable->False]}], SequenceForm[ 21, ") S = ", StringForm[ "{`1` , `2`}", 1, Times[ Power[ Log[ 2], -1], Log[ 3]]]], Editable->False], TraditionalForm]], "Print"], Cell[BoxData[ FormBox[ InterpretationBox[ RowBox[{ "22", "\[InvisibleSpace]", "\<\") S = \"\>", "\[InvisibleSpace]", InterpretationBox["\<\"{\\!\\(TraditionalForm\\`1\\/\[ExponentialE]\ \\)}\"\>", StringForm[ "{``}", Power[ E, -1]], Editable->False]}], SequenceForm[ 22, ") S = ", StringForm[ "{``}", Power[ E, -1]]], Editable->False], TraditionalForm]], "Print"], Cell[BoxData[ FormBox[ InterpretationBox[ RowBox[{ "23", "\[InvisibleSpace]", "\<\") S = \"\>", "\[InvisibleSpace]", InterpretationBox["\<\"{\\!\\(TraditionalForm\\`1\\/\[ExponentialE]\ \\^2\\) , \\!\\(TraditionalForm\\`\\@\[ExponentialE]\\%3\\)}\"\>", StringForm[ "{`1` , `2`}", Power[ E, -2], Power[ E, Rational[ 1, 3]]], Editable->False]}], SequenceForm[ 23, ") S = ", StringForm[ "{`1` , `2`}", Power[ E, -2], Power[ E, Rational[ 1, 3]]]], 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