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{"name": "Logarithms: Expanding Expressions 3", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Logarithms: Expanding Expressions 3", "tags": [], "metadata": {"description": "<p>Rewriting $\\log_{10}(\\sqrt{x})$ in terms of $\\log_{10}(a)$ and $\\log_{10}(b)$, where $a$, $b$ and $x$ are given.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Rewrite the following expression in terms of $\\simplify{log({a})}$ and $\\simplify{log({b})}$:</p>\n<p>\\[ \\simplify[fractionNumbers]{log(sqrt({a^3*b}))}.\\]</p>", "advice": "<p>To rewrite $\\simplify{log(sqrt({a^3*b}))}$ in terms of $\\simplify{log({a})}$ and $\\simplify{log({b})}$, we want to use the following logarithm rules:</p>\n<ul>\n<li>$\\simplify{log(a*b) = log(a)+log(b)}$;</li>\n<li>$\\simplify{log(a^n) = n*log(a)}$.</li>\n</ul>\n<p>Firstly, we want to express $\\var{a^3*b}$ in terms of $\\var{a}$ and $\\var{b}$:</p>\n<p>\\[ \\begin{split} \\var{a^3*b} &amp;\\,= \\var{a^3} \\times \\var{b} \\\\ &amp;\\,= \\var{a}^3 \\times \\var{b}. \\end{split} \\]</p>\n<p></p>\n<p>Hence,</p>\n<p>\\[ \\begin{split} \\simplify{log(sqrt({a^3*b}))} &amp;\\,=&nbsp; \\simplify{log({a^3*b}^(1/2))} \\\\ &amp;\\,=\\simplify[fractionNumbers]{(1/2)log({a^3*b})} \\\\ &amp;\\,=&nbsp; \\simplify[fractionNumbers,!collectNumbers]{(1/2)log({a}^3*{b})} \\\\ &amp;\\,= \\simplify[fractionNumbers,!collectNumbers]{(1/2)log({a}^3)+(1/2)log({b})} \\\\ &amp;\\,= \\simplify[fractionNumbers]{(3/2)log({a})+(1/2)log({b})} .\\end{split} \\]</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random([2,3])", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random([2,3] except a)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "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>[[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": "(3/2)log({a})+(1/2)log({b})", "answerSimplification": "fractionNumbers", "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/"}]}