// Numbas version: exam_results_page_options
{"name": "Solve an exponential equation", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Solve an exponential equation", "tags": [], "metadata": {"description": "<p>Solve exponential equation of the form \\[ a^{kx}=b^{kx+m}. \\]</p>", "licence": "Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International"}, "statement": "<p>Solve the equation \\[ \\var{base1}^{\\simplify{{k} x}} = \\var{base2}^{\\simplify{{k}x+{m}}} \\]</p>\n<p></p>", "advice": "<p>Since some powers are the same, begin&nbsp;by rewriting the equations as $\\var{base1}^{\\var{k}x} =&nbsp;\\var{base2}^{\\var{k}x}\\times&nbsp;&nbsp;\\var{base2}^{\\var{m}}$.</p>\n<p>Then group the terms with the same power to get: $\\left(\\frac{\\var{base1}}{\\var{base2}}\\right)^{\\var{k}x} =\\var{base2}^{\\var{m}}$.</p>\n<p>Take logs of both sides to obtain: $&nbsp;\\var{k}x = \\frac{\\ln(\\var{base2}^{\\var{m}})}{\\ln\\left(\\var{base1}/\\var{base2}\\right)}$.</p>\n<p>Therefore, $x=\\frac{\\ln(\\var{base2}^{\\var{m}})}{\\var{k}\\ln\\left(\\var{base1}/\\var{base2}\\right)}=${ (1/{k})*ln({base2}^{m})/ln({base1}/{base2}) }</p>", "rulesets": {}, "extensions": [], "variables": {"base1": {"name": "base1", "group": "Ungrouped variables", "definition": "random(2..10)", "description": "", "templateType": "anything"}, "k": {"name": "k", "group": "Ungrouped variables", "definition": "random(-10..10 except 0)", "description": "", "templateType": "anything"}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "random(-5..5 except [0,k,-k])", "description": "", "templateType": "anything"}, "base2": {"name": "base2", "group": "Ungrouped variables", "definition": "random(2..10 except base1)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["base1", "k", "m", "base2"], "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>", "stepsPenalty": 0, "steps": [{"type": "information", "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>Start by rewriting the equations as $\\var{base1}^{\\var{k}x} =&nbsp;\\var{base2}^{\\var{k}x}\\times&nbsp;&nbsp;\\var{base2}^{\\var{m}}$.</p>"}], "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "4", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "(1/{k})*ln({base2}^{m})/ln({base1}/{base2})", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": []}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Wan Mekwi", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4058/"}]}]}], "contributors": [{"name": "Wan Mekwi", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4058/"}]}