// Numbas version: exam_results_page_options
{"name": "Solving equations using log", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"preamble": {"js": "document.createElement('fraction');\ndocument.createElement('numerator');\ndocument.createElement('denominator');", "css": "fraction {\n  display: inline-block;\n  vertical-align: middle;\n}\nfraction > numerator, fraction > denominator {\n  float: left;\n  width: 100%;\n  text-align: center;\n  line-height: 2.5em;\n}\nfraction > numerator {\n  border-bottom: 1px solid;\n  padding-bottom: 5px;\n}\nfraction > denominator {\n  padding-top: 5px;\n}\nfraction input {\n    line-height: 1em;\n}\n\nfraction .part {\n    margin: 0;\n}\n\n.table-responsive, .fractiontable {\n  display:inline-block;\n}\n.fractiontable {\n  padding: 0; \n  border: 0;\n}\n\n.fractiontable  .tddenom \n{\n text-align: center;\n}\n\n.fractiontable .tdnum \n{\n  border-bottom: 1px solid black; \n  text-align: center;\n}\n\n\n.fractiontable tr {\n    height: 3em;\n}\n\n"}, "ungrouped_variables": ["a1", "a2", "b1", "b2", "c1", "c2", "d1", "d2", "e1", "e2", "e3", "e4", "e5", "e6", "f1", "f2", "f3", "f4", "f5", ""], "advice": "<p>To solve this type of question, we want to take logarithms of both sides of the equation in order to try to isolate x.</p>\n<p>For example, if we have $a^x = b$ then we proceed as above to get $x\\log{a} = \\log{b}$ and hence $x = \\frac{\\log{a}}{\\log{b}}$</p>\n<p>Laws of logarithms are useful&nbsp;here:</p>\n<p>$\\log{a} + \\log{b} = \\log{ab}$</p>\n<p>$\\log{a} - \\log{b} = \\log{\\frac{a}{b}}$</p>\n<p>$\\log{a^n} = n\\log{a}$</p>", "metadata": {"description": "", "licence": "Creative Commons Attribution 4.0 International"}, "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "parts": [{"customName": "", "sortAnswers": false, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "customMarkingAlgorithm": "", "scripts": {}, "showFeedbackIcon": true, "useCustomName": false, "variableReplacements": [], "prompt": "<p>$\\var{a1}^x = \\var{a2}$</p>\n<p>$x=$<fraction><numerator>$\\log_{10}$[[0]]</numerator><denominator>$\\log_{10}$[[1]]</denominator></fraction></p>\n<p>or</p>\n<p>$x=\\log_{\\var{a1}}$[[2]]</p>", "unitTests": [], "gaps": [{"customName": "", "answer": "{a2}", "variableReplacementStrategy": "originalfirst", "checkVariableNames": false, "checkingAccuracy": 0.001, "type": "jme", "showPreview": true, "customMarkingAlgorithm": "", "scripts": {}, "showFeedbackIcon": true, "useCustomName": false, "variableReplacements": [], "unitTests": [], "vsetRangePoints": 5, "vsetRange": [0, 1], "showCorrectAnswer": true, "marks": 1, "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "failureRate": 1, "valuegenerators": []}, {"customName": "", "answer": "{a1}", "variableReplacementStrategy": "originalfirst", "checkVariableNames": false, "checkingAccuracy": 0.001, "type": "jme", "showPreview": true, "customMarkingAlgorithm": "", "scripts": {}, "showFeedbackIcon": true, "useCustomName": false, "variableReplacements": [], "unitTests": [], "vsetRangePoints": 5, "vsetRange": [0, 1], "showCorrectAnswer": true, "marks": 1, "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "failureRate": 1, "valuegenerators": []}, {"customName": "", "answer": "{a2}", "variableReplacementStrategy": "originalfirst", "checkVariableNames": false, "checkingAccuracy": 0.001, "type": "jme", "showPreview": true, "customMarkingAlgorithm": "", "scripts": {}, "showFeedbackIcon": true, "useCustomName": false, "variableReplacements": [], "unitTests": [], "vsetRangePoints": 5, "vsetRange": [0, 1], "showCorrectAnswer": true, "marks": 1, "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "failureRate": 1, "valuegenerators": []}], "showCorrectAnswer": true, "marks": 0, "extendBaseMarkingAlgorithm": true}, {"customName": "", "sortAnswers": false, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "customMarkingAlgorithm": "", "scripts": {}, "showFeedbackIcon": true, "useCustomName": false, "variableReplacements": [], "prompt": "<p>$\\var{b1}^x=\\var{b2}$</p>\n<p>$x=$[[0]]</p>", "unitTests": [], "gaps": [{"variableReplacementStrategy": "originalfirst", "checkingType": "dp", "customMarkingAlgorithm": "", "type": "jme", "showFeedbackIcon": true, "useCustomName": false, "checkingAccuracy": 3, "vsetRangePoints": 5, "showCorrectAnswer": true, "marks": 1, "checkVariableNames": false, "valuegenerators": [], "customName": "", "answer": "log({b2})/log({b1})", "showPreview": true, "scripts": {}, "variableReplacements": [], "unitTests": [], "vsetRange": [0, 1], "extendBaseMarkingAlgorithm": true, "failureRate": 1, "answerSimplification": "all"}], "showCorrectAnswer": true, "marks": 0, "extendBaseMarkingAlgorithm": true}, {"customName": "", "sortAnswers": false, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "customMarkingAlgorithm": "", "scripts": {}, "showFeedbackIcon": true, "useCustomName": false, "variableReplacements": [], "prompt": "<p>$\\var{c1}^x = \\var{c2}$</p>\n<p>$x=$[[0]]</p>", "unitTests": [], "gaps": [{"variableReplacementStrategy": "originalfirst", "checkingType": "dp", "customMarkingAlgorithm": "", "type": "jme", "showFeedbackIcon": true, "useCustomName": false, "checkingAccuracy": 3, "vsetRangePoints": 5, "showCorrectAnswer": true, "marks": 1, "checkVariableNames": false, "valuegenerators": [], "customName": "", "answer": "log({c2})/log({c1})", "showPreview": true, "scripts": {}, "variableReplacements": [], "unitTests": [], "vsetRange": [0, 1], "extendBaseMarkingAlgorithm": true, "failureRate": 1, "answerSimplification": "all"}], "showCorrectAnswer": true, "marks": 0, "extendBaseMarkingAlgorithm": true}, {"customName": "", "sortAnswers": false, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "customMarkingAlgorithm": "", "scripts": {}, "showFeedbackIcon": true, "useCustomName": false, "variableReplacements": [], "prompt": "<p>$\\var{d1}^{x-1} = \\var{d2}^{x+1}$</p>\n<p>([[1]])$\\simplify{log({d1})}=$([[2]])$\\simplify{log({d2})}$</p>\n<p>Hence $x=$[[0]]</p>", "unitTests": [], "gaps": [{"customName": "", "answer": "log({d1}*{d2})/log({d1}/{d2})", "variableReplacementStrategy": "originalfirst", "checkVariableNames": false, "checkingAccuracy": 3, "type": "jme", "showPreview": true, "customMarkingAlgorithm": "", "scripts": {}, "showFeedbackIcon": true, "useCustomName": false, "variableReplacements": [], "unitTests": [], "vsetRangePoints": 5, "vsetRange": [0, 1], "showCorrectAnswer": true, "marks": 1, "extendBaseMarkingAlgorithm": true, "checkingType": "dp", "failureRate": 1, "valuegenerators": []}, {"customName": "", "answer": "x-1", "variableReplacementStrategy": "originalfirst", "checkVariableNames": false, "checkingAccuracy": 0.001, "type": "jme", "showPreview": true, "customMarkingAlgorithm": "", "scripts": {}, "showFeedbackIcon": true, "useCustomName": false, "variableReplacements": [], "unitTests": [], "vsetRangePoints": 5, "vsetRange": [0, 1], "showCorrectAnswer": true, "marks": 1, "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "failureRate": 1, "valuegenerators": [{"value": "", "name": "x"}]}, {"customName": "", "answer": "x+1", "variableReplacementStrategy": "originalfirst", "checkVariableNames": false, "checkingAccuracy": 0.001, "type": "jme", "showPreview": true, "customMarkingAlgorithm": "", "scripts": {}, "showFeedbackIcon": true, "useCustomName": false, "variableReplacements": [], "unitTests": [], "vsetRangePoints": 5, "vsetRange": [0, 1], "showCorrectAnswer": true, "marks": 1, "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "failureRate": 1, "valuegenerators": [{"value": "", "name": "x"}]}], "showCorrectAnswer": true, "marks": 0, "extendBaseMarkingAlgorithm": true}, {"customName": "", "sortAnswers": false, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "customMarkingAlgorithm": "", "scripts": {}, "showFeedbackIcon": true, "useCustomName": false, "variableReplacements": [], "prompt": "<p>$\\var{e1}^{\\simplify{{e3}x-{e4}}} = \\var{e2}^{\\simplify{{e5}x+{e6}}}$</p>\n<p>$x=$[[0]]</p>", "unitTests": [], "gaps": [{"variableReplacementStrategy": "originalfirst", "checkingType": "dp", "customMarkingAlgorithm": "", "type": "jme", "showFeedbackIcon": true, "useCustomName": false, "checkingAccuracy": 3, "vsetRangePoints": 5, "showCorrectAnswer": true, "marks": 1, "checkVariableNames": false, "valuegenerators": [], "customName": "", "answer": "({e6}log({e2})+{e4}log({e1}))/({e3}log({e1})-{e5}log({e2}))", "showPreview": true, "scripts": {}, "variableReplacements": [], "unitTests": [], "vsetRange": [0, 1], "extendBaseMarkingAlgorithm": true, "failureRate": 1, "answerSimplification": "all"}], "showCorrectAnswer": true, "marks": 0, "extendBaseMarkingAlgorithm": true}, {"customName": "", "sortAnswers": false, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "customMarkingAlgorithm": "", "scripts": {}, "showFeedbackIcon": true, "useCustomName": false, "variableReplacements": [], "prompt": "<p>$\\var{f1}^{\\simplify{{f3}x-{f4}}} = \\var{f2}^{\\simplify{{f5}x}}$</p>\n<p>$x=$[[0]]</p>", "unitTests": [], "gaps": [{"variableReplacementStrategy": "originalfirst", "checkingType": "dp", "customMarkingAlgorithm": "", "type": "jme", "showFeedbackIcon": true, "useCustomName": false, "checkingAccuracy": 3, "vsetRangePoints": 5, "showCorrectAnswer": true, "marks": 1, "checkVariableNames": false, "valuegenerators": [], "customName": "", "answer": "({f4}log({f1}))/({f3}log({f1})-{f5}log({f2}))", "showPreview": true, "scripts": {}, "variableReplacements": [], "unitTests": [], "vsetRange": [0, 1], "extendBaseMarkingAlgorithm": true, "failureRate": 1, "answerSimplification": "all"}], "showCorrectAnswer": true, "marks": 0, "extendBaseMarkingAlgorithm": true}], "tags": [], "statement": "<p>Solve each of the following equations for $x$, giving your answers in terms of base 10 logarithms OR to 3 decimal places.</p>\n<p>Remember to write $\\simplify{log(a)}$ you must type loga with brackets around the a.</p>", "variables": {"": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(-2..2 except 0)", "name": ""}, "d1": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(2..9)", "name": "d1"}, "b1": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(2..9 except a1)", "name": "b1"}, "c2": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(2..19 except c1 except b2 except a2)", "name": "c2"}, "f5": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(1..3 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"description": "", "group": "Ungrouped variables", "definition": "random(-2..2 except 0)", "name": "e4"}}, "rulesets": {}, "functions": {}, "extensions": [], "name": "Solving equations using log", "contributors": [{"name": "joshua boddy", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/557/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}]}], "contributors": [{"name": "joshua boddy", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/557/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}