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{"name": "Logarithms: Solving Logarithmic Equations 7 - Sum of Logarithms", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Logarithms: Solving Logarithmic Equations 7 - Sum of Logarithms", "tags": [], "metadata": {"description": "<p>Solving $\\log(y)+\\log(x)=\\frac{1}{n}\\log(ay^n)$ for $x$, where $a$ and $n$ are positive integers.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Solve for $x$:</p>\n<p>\\[ \\log(y)+\\log(x)=\\frac{1}{\\var{n}}\\log(\\var{a^n}y^\\var{n}).\\]</p>", "advice": "<p>To solve for $x$, we want to make use of the following logarithm rules:</p>\n<ul>\n<li>$\\log(a)+\\log(b) = \\log(ab)$;</li>\n<li>$\\log(a^n)=n \\log(a)$.</li>\n</ul>\n<p>Therefore,&nbsp;</p>\n<p>\\[ \\begin{split} &amp;\\log(y)+&amp;\\log(x) &amp;\\,= \\frac{1}{\\var{n}} \\log(\\var{a^n}y^\\var{n}) \\\\ &amp;\\implies &amp;\\log(yx) &amp;\\,= \\log\\left((\\var{a^n}y^\\var{n})^\\frac{1}{\\var{n}})\\right) \\\\ &amp;\\implies &amp;\\log(yx) &amp;\\,= \\log(\\var{a}y) \\end{split} \\]</p>\n<p>Recall that if $\\log(a) = \\log(b)$, then $a=b$.</p>\n<p>Hence,</p>\n<p>\\[ \\begin{split} &nbsp;\\log(yx) &amp;\\,= \\log(\\var{a}y) \\\\ \\implies \\quad yx &amp;\\,= \\var{a}y \\\\ \\implies \\quad x &amp;\\,= \\var{a}. \\end{split} \\]</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2..6)", "description": "", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "if(a>3,2,random([2,3]))", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "n"], "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": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{a}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}]}]}], "contributors": [{"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}]}