// Numbas version: exam_results_page_options
{"name": "Blathnaid's copy of Simon's copy of Q3 Indices; Solve for x", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"metadata": {"description": "<p>Indices</p>\n<p>rebelmaths</p>", "licence": "Creative Commons Attribution 4.0 International"}, "tags": [], "ungrouped_variables": ["num11", "num12", "num13", "nn1", "p1", "ans1", "num21", "num22", "p21", "p22", "nn21", "nn22", "ans2", "p31", "p32", "num31", "num32", "ans3", "p41", "p42", "num41", "num42", "num43", "ans4", "nn31", "nn32", "nn41", "nn42"], "rulesets": {}, "statement": "<p>Solve for $x$ in each of the following:</p>\n<p><em>Enter answer as a whole number or fraction</em></p>", "variablesTest": {"condition": "ans1<100 and ans2<100 and ans3<100 and ans4<100", "maxRuns": 100}, "advice": "<p><strong>(a)</strong></p>\n<p><strong></strong>$2^{\\var{num11}x-\\var{num12}} = \\var{nn1}^{x+\\var{num13}}$</p>\n<p>$2^{\\var{num11}x-\\var{num12}}= (2^\\var{log({nn1},2)})^{x+\\var{num13}}$</p>\n<p>$2^{\\var{num11}x-\\var{num12}}= 2^{\\var{log({nn1},2)}(x+\\var{num13})}$</p>\n<p>$\\implies&nbsp;\\var{num11}x-\\var{num12} =&nbsp;\\var{log({nn1},2)}(x+\\var{num13})$</p>\n<p>$\\implies&nbsp;\\var{num11}x-\\var{num12} =&nbsp;\\var{log({nn1},2)}x +\\var{log({nn1},2)*num13}$</p>\n<p>$\\implies&nbsp;\\var{num11-log({nn1},2)}x =&nbsp;\\var{num12+log({nn1},2)*num13}$</p>\n<p>$\\implies x =&nbsp;\\var{(num12+log({nn1},2)*num13)/(num11-log({nn1},2))}$</p>\n<p></p>\n<p><strong>(b)</strong></p>\n<p><strong></strong></p>\n<p>$(3^\\var{log({nn21},3)})^{x-\\var{num21}} = (3^\\var{log({nn22},3)})^{\\var{num22} - x}$</p>\n<p>$3^{\\var{log({nn21},3)}({x-\\var{num21}})} =3^{\\var{log({nn22},3)}({\\var{num22} - x})}$</p>\n<p>$\\implies&nbsp;\\var{log({nn21},3)}({x-\\var{num21}}) ={\\var{log({nn22},3)}({\\var{num22} - x})} $</p>\n<p>$\\implies&nbsp;\\var{log({nn21},3)} x-\\var{log({nn21},3)*num21} =\\var{log({nn22},3)*{num22}} -\\var{log({nn22},3)} x $</p>\n<p>$\\implies&nbsp;\\var{log({nn21},3)+log({nn22},3)} x =\\var{log({nn22},3)*{num22} +log({nn21},3)*num21} $</p>\n<p>$\\implies&nbsp; x =\\var{(log({nn22},3)*{num22}+log({nn21},3)*num21)/(log({nn21},3)+log({nn22},3))}&nbsp;$</p>\n<p></p>\n<p></p>\n<p><strong>(c)</strong></p>\n<p>$\\var{nn31}^{\\var{num31}x+\\var{num32}} = \\frac{1}{\\var{nn32}}$</p>\n<p>$(2^\\var{log(nn31,2)})^{\\var{num31}x+\\var{num32}}= \\var{2}^{-\\var{log({nn32},2)}}$</p>\n<p>$2^{\\var{log(nn31,2)}({\\var{num31}x+\\var{num32}})}= \\var{2}^{-\\var{log({nn32},2)}}$</p>\n<p>$\\implies \\var{log(nn31,2)}({\\var{num31}x+\\var{num32}})= -\\var{log({nn32},2)}$</p>\n<p>$\\implies&nbsp; \\var{log(nn31,2)*{num31}}x+\\var{log(nn31,2)*num32}=-\\var{log({nn32},2)}$</p>\n<p>$\\implies x=\\frac{-\\var{log({nn32},2)}-\\var{log(nn31,2)*num32}}{\\var{log(nn31,2)*{num31}}}=\\var{(-log({nn32},2)-log(nn31,2)*num32)/(log(nn31,2)*{num31})}$</p>\n<p></p>\n<p></p>\n<p><strong>(d)</strong></p>\n<p>$\\var{nn41}^{\\var{num41}x-\\var{num42}} = \\var{nn42}^{\\var{num43} - x}$</p>\n<p>$(5^\\var{log({nn41},5)})^{\\var{num41}x-\\var{num42}} = (5^\\var{log({nn42},5)})^{\\var{num43} - x}$</p>\n<p>$5^{\\var{log({nn41},5)}({\\var{num41}x-\\var{num42}})} =5^{\\var{log({nn42},5)}({\\var{num43} - x})}$</p>\n<p>$\\implies&nbsp;\\var{log({nn41},5)}({\\var{num41}x-\\var{num42}}) ={\\var{log({nn42},5)}({\\var{num43} - x})} $</p>\n<p>$\\implies&nbsp;\\var{log({nn41},5)*num41} x-\\var{log({nn41},5)*num42} =\\var{log({nn42},5)*{num43}} -\\var{log({nn42},5)} x $</p>\n<p>$\\implies&nbsp;\\var{log({nn41},5)*num41+log({nn42},5)} x =\\var{log({nn42},5)*{num43} +log({nn41},5)*num42} $</p>\n<p>$\\implies&nbsp; x =\\var{(log({nn42},5)*{num43}+log({nn41},5)*num42)/(log({nn41},5)*num41+log({nn42},5))}&nbsp;$</p>\n<p></p>", "extensions": [], "name": "Blathnaid's copy of Simon's copy of Q3 Indices; Solve for x", "preamble": {"css": "", "js": ""}, "functions": {}, "variables": {"nn32": {"description": "", "templateType": "anything", "name": "nn32", "definition": "2^p32", "group": "Ungrouped variables"}, "p1": {"description": "", "templateType": "anything", "name": "p1", "definition": "random(2..3)", "group": "Ungrouped variables"}, "ans2": {"description": "", "templateType": "anything", "name": "ans2", "definition": "(num22*p22+num21*p21)/(p21+p22)", "group": "Ungrouped variables"}, "num13": {"description": "", "templateType": "anything", "name": "num13", "definition": "random(1..2)", "group": "Ungrouped variables"}, "num11": {"description": "", "templateType": "anything", "name": "num11", "definition": "random(3..6)", "group": "Ungrouped variables"}, "num31": {"description": "", "templateType": "anything", "name": "num31", "definition": "random(3..6)", "group": "Ungrouped variables"}, "num42": {"description": "", "templateType": "anything", "name": "num42", "definition": "random(3..6 except num41)", "group": "Ungrouped variables"}, "ans1": {"description": "", "templateType": "anything", "name": "ans1", "definition": "(num12+num13*p1)/(num11-p1)", "group": "Ungrouped variables"}, "num21": {"description": "", "templateType": "anything", "name": "num21", "definition": "random(3..6)", "group": "Ungrouped variables"}, "p41": {"description": "", "templateType": "anything", "name": "p41", "definition": "random(2..4)", "group": "Ungrouped variables"}, "ans3": {"description": "", "templateType": "anything", "name": "ans3", "definition": "((p32*-1)-num32*p31)/(p31*num31)", "group": "Ungrouped variables"}, "nn22": {"description": "", "templateType": "anything", "name": "nn22", "definition": "3^(p22)", "group": "Ungrouped variables"}, "num32": {"description": "", "templateType": "anything", "name": "num32", "definition": "random(3..6 except num31)", "group": "Ungrouped variables"}, "ans4": {"description": "", "templateType": "anything", "name": "ans4", "definition": "((p42*num43)+num42*p41)/((p41*num41)+p42)", "group": "Ungrouped variables"}, "nn41": {"description": "", "templateType": "anything", "name": "nn41", "definition": "5^p41", "group": "Ungrouped variables"}, "nn1": {"description": "", "templateType": "anything", "name": "nn1", "definition": "2^(p1)", "group": "Ungrouped variables"}, "nn42": {"description": "", "templateType": "anything", "name": "nn42", "definition": "5^p42", "group": "Ungrouped variables"}, "num12": {"description": "", "templateType": "anything", "name": "num12", "definition": "random(3..5 except num11)", "group": "Ungrouped variables"}, "num22": {"description": "", "templateType": "anything", "name": "num22", "definition": "random(2..6)", "group": "Ungrouped variables"}, "nn21": {"description": "", "templateType": "anything", "name": "nn21", "definition": "3^(p21)", "group": "Ungrouped variables"}, "p31": {"description": "", "templateType": "anything", "name": "p31", "definition": "random(1..3)", "group": "Ungrouped variables"}, "p22": {"description": "", "templateType": "anything", "name": "p22", "definition": "random(2..3 except p21)", "group": "Ungrouped variables"}, "nn31": {"description": "", "templateType": "anything", "name": "nn31", "definition": "2^p31", "group": "Ungrouped variables"}, "num43": {"description": "", "templateType": "anything", "name": "num43", "definition": "random(3..6 except [num41,num42])", "group": "Ungrouped variables"}, "p42": {"description": "", "templateType": "anything", "name": "p42", "definition": "random(2..3 except p41)", "group": "Ungrouped variables"}, "num41": {"description": "", "templateType": "anything", "name": "num41", "definition": "random(3..6)", "group": "Ungrouped variables"}, "p32": {"description": "", "templateType": "anything", "name": "p32", "definition": "random(1..3 except p31)", "group": "Ungrouped variables"}, "p21": {"description": "", "templateType": "anything", "name": "p21", "definition": "random(2..3)", "group": "Ungrouped variables"}}, "variable_groups": [], "parts": [{"showCorrectAnswer": true, "marks": 0, "scripts": {}, "type": "gapfill", "variableReplacements": [], "unitTests": [], "gaps": [{"correctAnswerFraction": true, "showCorrectAnswer": true, "mustBeReduced": false, "marks": 1, "scripts": {}, "allowFractions": true, "type": "numberentry", "variableReplacements": [], "unitTests": [], "notationStyles": ["plain", "en", "si-en"], "maxValue": "{ans1}", "correctAnswerStyle": "plain", "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "mustBeReducedPC": 0, "variableReplacementStrategy": "originalfirst", "minValue": "{ans1}", "customMarkingAlgorithm": ""}], "sortAnswers": false, "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "prompt": "<p>$2^{\\var{num11}x-\\var{num12}} = \\var{nn1}^{x+\\var{num13}}$</p>\n<p>$x =$&nbsp;[[0]]</p>", "customMarkingAlgorithm": ""}, {"showCorrectAnswer": true, "marks": 0, "scripts": {}, "type": "gapfill", "variableReplacements": [], "unitTests": [], "gaps": [{"correctAnswerFraction": true, "showCorrectAnswer": true, "mustBeReduced": false, "marks": 1, "scripts": {}, "allowFractions": true, "type": "numberentry", "variableReplacements": [], "unitTests": [], "notationStyles": ["plain", "en", "si-en"], "maxValue": "{ans2}", "correctAnswerStyle": "plain", "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "mustBeReducedPC": 0, "variableReplacementStrategy": "originalfirst", "minValue": "{ans2}", "customMarkingAlgorithm": ""}], "sortAnswers": false, "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "prompt": "<p>$\\var{nn21}^{x-\\var{num21}} = \\var{nn22}^{\\var{num22} - x}$</p>\n<p>$x=$&nbsp;[[0]]</p>", "customMarkingAlgorithm": ""}], "type": "question", "contributors": [{"name": "Blathnaid Sheridan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/447/"}, {"name": "TEAME CIT", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/591/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}]}], "contributors": [{"name": "Blathnaid Sheridan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/447/"}, {"name": "TEAME CIT", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/591/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}