(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 11.3' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] NotebookDataLength[ 65570, 1659] NotebookOptionsPosition[ 55545, 1528] NotebookOutlinePosition[ 56007, 1546] CellTagsIndexPosition[ 55964, 1543] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["T08: Exponential & Logarithmic Functions", "Title", CellChangeTimes->{{3.744615012185155*^9, 3.7446150291633053`*^9}}, TextAlignment->Center,ExpressionUUID->"956406c9-f929-4314-b5de-5dbb1a72ebaa"], Cell[TextData[{ "This notebook covers the Mathematica commands needs for this topic.\n\n\ There are 5 questions in this notebook.\n\nQuestion 1 - Simplify (be careful, \ sometimes, Mathematica will not fully simplify it for you). 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To specify \ others bases, please look at how c) is expressed below.", "Text", CellChangeTimes -> {{3.744615566566184*^9, 3.744615674132224*^9}, 3.744615806901223*^9}], Cell[ CellGroupData[{ Cell[ BoxData[ RowBox[{ RowBox[{"Log", "[", RowBox[{"3", ",", "18"}], "]"}], "-", RowBox[{"Log", "[", RowBox[{"3", ",", "6"}], "]"}]}]], "Input", CellChangeTimes -> {{3.7446155555951138`*^9, 3.744615556198511*^9}, {3.744615676129471*^9, 3.744615708683228*^9}}, CellLabel -> "In[10]:="], Cell[ BoxData[ RowBox[{ RowBox[{"-", FractionBox[ RowBox[{"Log", "[", "6", "]"}], RowBox[{"Log", "[", "3", "]"}]]}], "+", FractionBox[ RowBox[{"Log", "[", "18", "]"}], RowBox[{"Log", "[", "3", "]"}]]}]], "Output", CellChangeTimes -> {{3.744615688230871*^9, 3.744615709391522*^9}}, CellLabel -> "Out[10]="]}, Open]], Cell[ "Mathematica converts it to a natural logarithm. Add in the //N to \ find the actual answer. Or add in a decimal next to the 6.", "Text", CellChangeTimes -> {{3.744615720931246*^9, 3.744615774411818*^9}}], Cell[ CellGroupData[{ Cell[ BoxData[ RowBox[{ RowBox[{ RowBox[{"Log", "[", RowBox[{"3", ",", "18"}], "]"}], "-", RowBox[{"Log", "[", RowBox[{"3", ",", "6"}], "]"}]}], "//", "N"}]], "Input", CellChangeTimes -> {{3.744615753264483*^9, 3.744615784612459*^9}}, CellLabel -> "In[13]:="], Cell[ BoxData["0.9999999999999996`"], "Output", CellChangeTimes -> {{3.744615764999309*^9, 3.744615785123624*^9}}, CellLabel -> "Out[13]="]}, Open]], Cell[ CellGroupData[{ Cell[ BoxData[ RowBox[{ RowBox[{"Log", "[", RowBox[{"3", ",", "18"}], "]"}], "-", RowBox[{"Log", "[", RowBox[{"3", ",", "6."}], "]"}]}]], "Input", CellChangeTimes -> {{3.744615794032958*^9, 3.744615798201552*^9}}, CellLabel -> "In[14]:="], Cell[ BoxData["0.9999999999999996`"], "Output", CellChangeTimes -> {3.7446157988621902`*^9}, CellLabel -> "Out[14]="]}, Open]]}, WindowTitle -> "hint", WindowSize -> {800, 600}, Magnification -> 1.25, StyleDefinitions -> "Creative/PastelColor.nb"], Evaluator->Automatic, FrameMargins->4, Method->"Queued"]],ExpressionUUID->"0094a400-d3b0-4d83-9266-3a0a72eda872"] }], "Text", CellChangeTimes->{{3.7446154518945093`*^9, 3.744615471101206*^9}, { 3.744615896652255*^9, 3.744615907290059*^9}},ExpressionUUID->"cb590928-e8be-4af0-b232-\ dd0cebb48ddd"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"Log", "[", RowBox[{"3", ",", "18"}], "]"}], "-", RowBox[{"Log", "[", RowBox[{"3", ",", "6"}], "]"}]}], "//", "N"}]], "Input", CellChangeTimes->{{3.744615482470697*^9, 3.7446155447770042`*^9}, { 3.745543090266871*^9, 3.745543108929284*^9}}, CellLabel-> "In[101]:=",ExpressionUUID->"d72b24d1-68bb-4c7b-8eb8-94b383bad754"], Cell[BoxData["0.9999999999999996`"], "Output", CellChangeTimes->{{3.7455431059274073`*^9, 3.745543109583846*^9}}, CellLabel-> "Out[101]=",ExpressionUUID->"af2ccec9-dca6-4524-9b88-b8760b6cbabf"] }, Open ]], Cell[TextData[{ "\nd) ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"3", SubscriptBox["log", "5"], "3"}], "-", RowBox[{"2", SubscriptBox["log", "5"], "2"}]}], TraditionalForm]],ExpressionUUID-> "a5281755-527b-40c2-a5e9-2f2ce4a2666f"] }], "Text", CellChangeTimes->{{3.744615915879607*^9, 3.744615917233636*^9}, 3.744615963550333*^9},ExpressionUUID->"395bed6b-e0a6-4559-91a2-\ 3ea857ecfab5"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"3", "*", RowBox[{"Log", "[", RowBox[{"5", ",", "3"}], "]"}]}], "-", RowBox[{"2", "*", RowBox[{"Log", "[", RowBox[{"5", ",", "2"}], "]"}]}]}], "//", "N"}]], "Input", CellChangeTimes->{{3.744615965407362*^9, 3.744615965905561*^9}, { 3.7455431307088614`*^9, 3.74554315699638*^9}}, CellLabel-> "In[103]:=",ExpressionUUID->"1043df10-96a7-4497-8580-823aecbf77e8"], Cell[BoxData["1.1864654673111699`"], "Output", CellChangeTimes->{{3.745543153614748*^9, 3.7455431579260597`*^9}}, CellLabel-> "Out[103]=",ExpressionUUID->"2dfef6ce-3c4d-4d34-a193-e64911e76032"] }, Open ]], Cell["", "Text",ExpressionUUID->"073699bd-81ec-4da4-830f-901616df4b62"], Cell["", "PageBreak", PageBreakBelow->True,ExpressionUUID->"e755df60-b84d-452b-8531-5ac75f03acf1"], Cell[TextData[{ "Question 2 - Evaluate\n\na) ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["3", "x"], "=", "81"}], TraditionalForm]],ExpressionUUID-> "7bbc5c06-b853-40e2-b892-aadac60c5ab4"] }], "Text", CellChangeTimes->{{3.7446160062344427`*^9, 3.744616046384481*^9}},ExpressionUUID->"5881f74e-29a4-4969-9c8e-\ bf9c78cf523f"], Cell[BoxData[""], "Input", CellChangeTimes->{{3.744616047760725*^9, 3.744616048231513*^9}},ExpressionUUID->"8cfb216d-d808-4101-b703-\ 6cf718ce7821"], Cell["\<\ Answer\ \>", "Text", CellChangeTimes->{{3.742498818130303*^9, 3.742498832880702*^9}},ExpressionUUID->"283aa6e4-34a0-4950-a860-\ ec173eaf49a1"], Cell["", "Text", Background->RGBColor[ 1, 0.85, 0.85],ExpressionUUID->"f0ac1273-7d91-45b2-b1fc-22f27016d158"], Cell[TextData[{ "\nb) ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["2", RowBox[{"2", "p"}]], "=", RowBox[{ RowBox[{"6", RowBox[{"(", SuperscriptBox["2", "p"], ")"}]}], "+", "16"}]}], TraditionalForm]], ExpressionUUID->"8ebd25c6-cd50-4650-b2a1-764a4c08ce03"] }], "Text", CellChangeTimes->{{3.744616083997636*^9, 3.744616086300921*^9}},ExpressionUUID->"ac2a3a89-9f5f-46bc-8319-\ a060497099e2"], Cell[BoxData[""], "Input", CellChangeTimes->{{3.744616089021082*^9, 3.74461608952043*^9}},ExpressionUUID->"c1ff3b2e-085d-458e-9902-\ f623d6453fea"], Cell["\<\ Answer\ \>", "Text", CellChangeTimes->{{3.742498818130303*^9, 3.742498832880702*^9}},ExpressionUUID->"c8627db4-8a47-488f-a63f-\ 1b98e5e3b066"], Cell["", "Text", Background->RGBColor[ 1, 0.85, 0.85],ExpressionUUID->"f8f05c70-c0fc-43f4-a5d1-fba2defa4ffc"], Cell[TextData[{ "\nc) ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["12", "x"], "=", "1200"}], TraditionalForm]], ExpressionUUID->"94e2e269-68f3-485a-b695-bf01f4b8f8f2"], ", to two decimal places." }], "Text", CellChangeTimes->{{3.744616126259452*^9, 3.744616160299801*^9}},ExpressionUUID->"98a1a4b1-d849-4f12-90c6-\ 9ba95600d1fc"], Cell[BoxData[""], "Input", CellChangeTimes->{{3.744616089021082*^9, 3.74461608952043*^9}},ExpressionUUID->"9227b723-77a4-49b3-956e-\ 0e29d51e7bfe"], Cell["\<\ Answer\ \>", "Text", CellChangeTimes->{{3.742498818130303*^9, 3.742498832880702*^9}},ExpressionUUID->"f5eeef84-d4dd-4681-a251-\ 6e4e51e583ff"], Cell["", "Text", Background->RGBColor[ 1, 0.85, 0.85],ExpressionUUID->"f2f48bc2-0547-4b6f-8824-b6b7c0eef42a"], Cell[TextData[{ "\nd) ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["log", "5"], "x"}], "=", "3.1"}], TraditionalForm]], ExpressionUUID->"0a3be211-4b12-4e56-ad79-0eb0fb6d85c7"], ", to three decimal places." }], "Text", CellChangeTimes->{{3.744616208634122*^9, 3.7446162178570843`*^9}},ExpressionUUID->"513ed9e4-508d-49a2-a9dd-\ e7bef6f06580"], Cell[BoxData[""], "Input", CellChangeTimes->{{3.744616089021082*^9, 3.74461608952043*^9}},ExpressionUUID->"5c9507c3-7d75-4d2f-911e-\ 27437e7ed69f"], Cell["\<\ Answer\ \>", "Text", CellChangeTimes->{{3.742498818130303*^9, 3.742498832880702*^9}},ExpressionUUID->"52961a21-e7e0-43f3-9b6e-\ a4f6d41b0925"], Cell["", "Text", Background->RGBColor[ 1, 0.85, 0.85],ExpressionUUID->"ae203f90-ea07-4f72-a0f7-27afb03dd3a6"], Cell["", "Text",ExpressionUUID->"1940c603-f6d5-45ac-ae5d-a2ae7ef5473d"], Cell["", "PageBreak", PageBreakBelow->True,ExpressionUUID->"faac62b1-50c0-43d4-8b29-1dda2284d56a"], Cell[TextData[{ "Question 3 - Consider the function ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"f", "(", "x", ")"}], "=", RowBox[{ RowBox[{ RowBox[{"-", "8"}], "\[Times]", SuperscriptBox["3", RowBox[{ RowBox[{"5", "x"}], "+", "2"}]]}], "+", "7"}]}], TraditionalForm]], FormatType->"TraditionalForm",ExpressionUUID-> "94e3b3aa-3b86-490e-bf96-453a064b4b87"], "\n\na) Find the coordinate of the y-intercept." }], "Text", CellChangeTimes->{{3.744616291896345*^9, 3.7446163742547607`*^9}, { 3.744616622701603*^9, 3.744616642278492*^9}},ExpressionUUID->"92d365a7-2ce3-4da2-84d2-\ 21687618ad99"], Cell[BoxData[""], "Input", CellChangeTimes->{{3.744616089021082*^9, 3.74461608952043*^9}},ExpressionUUID->"bf851d77-994f-4425-8968-\ cca3e25cccd3"], Cell["\<\ Answer\ \>", "Text", CellChangeTimes->{{3.742498818130303*^9, 3.742498832880702*^9}},ExpressionUUID->"ddbf3350-77ee-40ca-962d-\ e4de2e3b19a8"], Cell["", "Text", Background->RGBColor[ 1, 0.85, 0.85],ExpressionUUID->"71df0d3c-7214-4a22-8b02-a274d491e41e"], Cell["\<\ b) Find the coordinate of the x-intercept.\ \>", "Text", CellChangeTimes->{{3.744616392460936*^9, 3.7446164050717297`*^9}},ExpressionUUID->"781a3751-5e70-4a8b-9159-\ 4b2832e0eb50"], Cell[BoxData[""], "Input", CellChangeTimes->{{3.744616089021082*^9, 3.74461608952043*^9}},ExpressionUUID->"b8578137-72a3-4f7a-9497-\ 75d0259fcb0d"], Cell["\<\ Answer\ \>", "Text", CellChangeTimes->{{3.742498818130303*^9, 3.742498832880702*^9}},ExpressionUUID->"1dd282df-783d-4732-b051-\ f31c644318d4"], Cell["", "Text", Background->RGBColor[ 1, 0.85, 0.85],ExpressionUUID->"f94c6ce0-aee8-4982-9cb4-0e8bb6e15755"], Cell[TextData[{ "\nc) State the transformation which take ", Cell[BoxData[ FormBox[ RowBox[{"y", "=", SuperscriptBox["3", "x"]}], TraditionalForm]], FormatType->"TraditionalForm",ExpressionUUID-> "0a7086ca-7d36-40e4-8104-b941fa2d6b11"], " to ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"f", "(", "x", ")"}], "=", RowBox[{ RowBox[{ RowBox[{"-", "8"}], "\[Times]", SuperscriptBox["3", RowBox[{ RowBox[{"5", "x"}], "+", "2"}]]}], "+", "7"}]}], TraditionalForm]], ExpressionUUID->"8175f384-86e8-4aec-9f09-fc0f4af8008f"], "." }], "Text", CellChangeTimes->{{3.744616410548526*^9, 3.744616459270211*^9}, { 3.744616650593*^9, 3.744616669826478*^9}},ExpressionUUID->"852b3cb7-816e-4994-bd9a-\ e719f09ddfba"], Cell[BoxData[""], "Input", CellChangeTimes->{{3.744616089021082*^9, 3.74461608952043*^9}},ExpressionUUID->"075e1172-2312-4139-bc1b-\ da131a35377c"], Cell["\<\ Answer\ \>", "Text", CellChangeTimes->{{3.742498818130303*^9, 3.742498832880702*^9}},ExpressionUUID->"1731d92a-b3a1-46b8-bb27-\ b4d433d84f0f"], Cell["", "Text", Background->RGBColor[ 1, 0.85, 0.85],ExpressionUUID->"84b97044-caf6-4558-a864-ee0c977e932e"], Cell["\<\ d) What is the equation of the asymptote?\ \>", "Text", CellChangeTimes->{{3.7446164719241734`*^9, 3.7446164785539703`*^9}},ExpressionUUID->"4e10671f-b8f5-4c70-8dfc-\ bf72b4031052"], Cell[BoxData[""], "Input", CellChangeTimes->{{3.744616089021082*^9, 3.74461608952043*^9}},ExpressionUUID->"20f57a86-543a-4c40-b196-\ f238a593b8e9"], Cell["\<\ Answer\ \>", "Text", CellChangeTimes->{{3.742498818130303*^9, 3.742498832880702*^9}},ExpressionUUID->"fe6bccea-5fb3-4e6d-a1d7-\ dde1a05f7775"], Cell["", "Text", Background->RGBColor[ 1, 0.85, 0.85],ExpressionUUID->"163ca57a-b3b9-4783-819b-bfabb725e088"], Cell[TextData[{ "\ne) Plot the graph of ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"f", "(", "x", ")"}], "=", RowBox[{ RowBox[{ RowBox[{"-", "8"}], "\[Times]", SuperscriptBox["3", RowBox[{ RowBox[{"5", "x"}], "+", "2"}]]}], "+", "7"}]}], TraditionalForm]], ExpressionUUID->"0757d32e-669f-428c-861a-006aaae416ef"], " and label all key information with coordinates and asymptote(s) with their \ equation." }], "Text", CellChangeTimes->{{3.744616485832534*^9, 3.744616535910631*^9}, { 3.744616676850523*^9, 3.744616676850614*^9}},ExpressionUUID->"da94772c-a68c-4340-aecf-\ 15bbf35bd017"], Cell[BoxData[""], "Input", CellChangeTimes->{{3.744616089021082*^9, 3.74461608952043*^9}},ExpressionUUID->"a8b03451-1a73-4075-b3c5-\ 367b4a8dd023"], Cell["\<\ Answer\ \>", "Text", CellChangeTimes->{{3.742498818130303*^9, 3.742498832880702*^9}},ExpressionUUID->"02d85fda-4f58-4900-ab2e-\ aa792bb89db4"], Cell["", "Text", Background->RGBColor[ 1, 0.85, 0.85],ExpressionUUID->"6765bbd7-ae6b-4a74-abd0-dbb624f0c677"], Cell["", "Text",ExpressionUUID->"ec2ce3a5-19b2-4ecb-be1b-98ca58006028"], Cell["", "PageBreak", PageBreakBelow->True,ExpressionUUID->"2b56a3ee-ab43-48c9-8541-41bbaa901973"], Cell[TextData[{ "Question 4 - 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