// Numbas version: exam_results_page_options
{"name": "Equations involving logs", "metadata": {"description": "<p>Solve equations involving logs and exponential functions, by using inverse operations.</p>", "licence": "Creative Commons Attribution-ShareAlike 4.0 International"}, "duration": 0, "percentPass": 0, "showQuestionGroupNames": false, "shuffleQuestionGroups": false, "showstudentname": true, "question_groups": [{"name": "Group", "pickingStrategy": "all-shuffled", "pickQuestions": 1, "questionNames": ["", "", "", "", ""], "variable_overrides": [[], [], [], [], []], "questions": [{"name": "Exponential equation 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Martin Jones", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/145/"}], "tags": [], "metadata": {"description": "<p>Solve equation involving exponential function.</p>", "licence": "Creative Commons Attribution-ShareAlike 4.0 International"}, "statement": "<p>Solve the equation</p>\n<p>\\[\\simplify{{A}^(x+{c})/{B}}=\\var{D}\\]</p>", "advice": "<p>Multiply by $\\var{B}$:</p>\n<p>\\[\\simplify{{A}^(x+{c})}=\\var{B}\\times\\var{D}=\\var{B*D}\\]</p>\n<p>Take logarithm base $\\var{A}$:</p>\n<p>\\[\\simplify{x+{c}}=\\log_\\var{A}(\\var{B*D})=\\var{log(B*D)/log(A)}\\]</p>\n<p>Add/subtract $\\var{c}$:</p>\n<p>\\[x=\\var{x}\\]</p>\n<p>Rounding to 2 decimal places gives $x=\\var{precround(x,2)}$</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"A": {"name": "A", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "templateType": "anything", "can_override": false}, "B": {"name": "B", "group": "Ungrouped variables", "definition": "random(2..10)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything", "can_override": false}, "D": {"name": "D", "group": "Ungrouped variables", "definition": "random(10..50 #0.1)", "description": "", "templateType": "anything", "can_override": false}, "x": {"name": "x", "group": "Ungrouped variables", "definition": "log(B*D)/log(A)-c", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["A", "B", "c", "D", "x"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": 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"always", "penaltyVisibility": "always"}, {"name": "Exponential equation 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Martin Jones", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/145/"}], "tags": [], "metadata": {"description": "<p>Solve equation involving exponential function</p>", "licence": "Creative Commons Attribution-ShareAlike 4.0 International"}, "statement": "<p>Solve the equation</p>\n<p>\\[\\simplify{sqrt(exp({A}+x))}=\\var{D}\\]</p>", "advice": "<p>Starting from the equation:</p>\n<p>\\[\\simplify{sqrt(exp({A}+x))}=\\var{D}\\]</p>\n<p>Square:</p>\n<p>\\[\\simplify{exp({A}+x)}=\\var{D}^2=\\var{D^2}\\]</p>\n<p>Take natural log:</p>\n<p>\\[\\simplify{{A}+x}=\\ln(\\var{D^2})=\\var{ln(D^2)}\\]</p>\n<p>Subtract $\\var{A}$:</p>\n<p>\\[x=\\var{ln(D^2)}-\\var{A}=\\var{x}\\]</p>\n<p>Rounding to 2 decimal places gives $x=\\var{precround(x,2)}$</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"A": {"name": "A", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "templateType": "anything", "can_override": false}, "D": {"name": "D", "group": "Ungrouped variables", "definition": "random(10..50 #0.1)", "description": "", "templateType": "anything", "can_override": false}, "x": {"name": "x", "group": "Ungrouped variables", "definition": "ln(D^2)-A", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["A", "D", "x"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, 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"showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Exponential equation 3", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Martin Jones", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/145/"}], "tags": [], "metadata": {"description": "<p>Solve equation involving exponential function.</p>", "licence": "Creative Commons Attribution-ShareAlike 4.0 International"}, "statement": "<p>Solve the equation</p>\n<p>\\[\\simplify{{A}/10^({B}*x-{C})}=\\var{D}\\]</p>", "advice": "<p>Starting from the equation:</p>\n<p>\\[\\simplify{{A}/10^({B}*x-{C})}=\\var{D}\\]</p>\n<p>Divide into $\\var{A}$:</p>\n<p>\\[\\simplify{10^({B}*x-{C})}=\\frac{\\var{A}}{\\var{D}}=\\var{A/D}\\]</p>\n<p>Take log base 10:</p>\n<p>\\[\\simplify{{B}*x-{C}}=\\log(\\var{A/D})=\\var{log(A/D)}\\]</p>\n<p>Add $\\var{C}$:</p>\n<p>\\[\\simplify{{B}*x}=\\var{log(A/D)}+\\var{C}=\\var{log(A/D)+C}\\]</p>\n<p>Divide by $\\var{B}$:</p>\n<p>\\[x=\\frac{\\var{log(A/D)+C}}{\\var{B}}=\\var{x}\\]</p>\n<p>Rounding to 2 decimal places gives $x=\\var{precround(x,2)}$</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"A": {"name": "A", "group": "Ungrouped variables", "definition": "D*random(0.5..3 #0.1)", "description": "", "templateType": "anything", "can_override": false}, "D": {"name": "D", "group": "Ungrouped variables", "definition": "random(10..50)", "description": "", "templateType": "anything", "can_override": false}, "x": {"name": "x", "group": "Ungrouped variables", "definition": "(log(A/D)+C)/b", "description": "", "templateType": "anything", "can_override": false}, "C": {"name": "C", "group": "Ungrouped variables", "definition": "random(5..20)", "description": "", "templateType": "anything", "can_override": false}, "B": {"name": "B", "group": "Ungrouped variables", "definition": "random(2..6)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["A", "D", "x", "C", "B"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$x=\\;$[[0]]</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{x}", "maxValue": "{x}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "precisionType": "dp", "precision": "2", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": true, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Log equation 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Martin Jones", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/145/"}], "tags": [], "metadata": {"description": "<p>Solve equation involving logarithmic function.</p>", "licence": "Creative Commons Attribution-ShareAlike 4.0 International"}, "statement": "<p>Solve the equation</p>\n<p>\\[\\simplify{{A}-ln((x+{B})/{C})}=\\var{D}\\]</p>", "advice": "<p>Starting from the equation:</p>\n<p>\\[\\simplify{{A}-ln((x+{B})/{C})}=\\var{D}\\]</p>\n<p>Subtract from $\\var{A}$:</p>\n<p>\\[\\simplify{ln((x+{B})/{C})}=\\var{A}-\\var{D}=\\var{A-D}\\]</p>\n<p>Take $e$ to the power:</p>\n<p>\\[\\simplify{(x+{B})/{C}}=e^\\var{A-D}=\\var{exp(A-D)}\\]</p>\n<p>Multiply by $\\var{C}$:</p>\n<p>\\[\\simplify{x+{B}}=\\var{C}\\times\\var{exp(A-D)}=\\var{C*exp(A-D)}\\]</p>\n<p>Add/subtract $\\var{abs(B)}$:</p>\n<p>\\[x=\\var{x}\\]</p>\n<p>Rounding to 2 decimal places gives $x=\\var{precround(x,2)}$.</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"C": {"name": "C", "group": "Ungrouped variables", "definition": "random(2..10)", "description": "", "templateType": "anything", "can_override": false}, "D": {"name": "D", "group": "Ungrouped variables", "definition": "A-random(1..4 #0.1)", "description": "", "templateType": "anything", "can_override": false}, "B": {"name": "B", "group": "Ungrouped variables", "definition": "random(-8..8 except 0)", "description": "", "templateType": "anything", "can_override": false}, "A": {"name": "A", "group": "Ungrouped variables", "definition": "random(5..10 #0.1)", "description": "", "templateType": "anything", "can_override": false}, "x": {"name": "x", "group": "Ungrouped variables", "definition": "C*exp(A-D)-B", "description": "", "templateType": "anything", 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false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "x", "maxValue": "x", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "precisionType": "dp", "precision": "2", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": true, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Log equation 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Martin Jones", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/145/"}], "tags": [], "metadata": {"description": "<p>Solve equation involving logarithmic function.</p>", "licence": "Creative Commons Attribution-ShareAlike 4.0 International"}, "statement": "<p>Solve the equation</p>\n<p>\\[\\simplify{{B}*log(({A}-x)^2)}=\\var{D}\\]</p>", "advice": "<p>Starting from the equation</p>\n<p>\\[\\simplify{{B}*log(({A}-x)^2)}=\\var{D}\\]</p>\n<p>Divide by $\\var{B}$:</p>\n<p>\\[\\simplify{log(({A}-x)^2)}=\\frac{\\var{D}}{\\var{B}}=\\var{D/B}\\]</p>\n<p>Take $10$ to the power:</p>\n<p>\\[\\simplify{({A}-x)^2}=10^\\var{D/B}=\\var{10^(D/B)}\\]</p>\n<p>Square root:</p>\n<p>\\[\\simplify{{A}-x}=\\pm\\sqrt{\\var{10^(D/B)}}=\\pm\\var{sqrt(10^(D/B))}\\]</p>\n<p>Subtract from $\\var{A}$:</p>\n<p>\\[x=\\var{A}\\mp\\var{sqrt(10^(D/B))}=\\var{x}\\text{ or }\\var{x2}\\]</p>\n<p>Rounding to 2 decimal places gives $x=\\var{precround(x,2)}$ or $x=\\var{precround(x2,2)}$</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"C": {"name": "C", 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