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{"name": "Logarithms: Simplifying Expressions 3", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Logarithms: Simplifying Expressions 3", "tags": [], "metadata": {"description": "<p>Rewriting expressions of the form $n \\log(a)\\pm m \\log(b) \\pm p \\log(c)$ as a single logarithm.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Rewrite the following expression as a single logarithm:</p>\n<p>\\[\\simplify{{n1}log({a})+{n2}log({b})+{n3}log({c})}.\\]</p>", "advice": "<p>{advice}</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"n1": {"name": "n1", "group": "Ungrouped variables", "definition": "random(2..3)", "description": "", "templateType": "anything", "can_override": false}, "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..5 except a)", "description": "", "templateType": "anything", "can_override": false}, "advice": {"name": "advice", "group": "Ungrouped variables", "definition": "if(n2>0 and n3>0,'{advice1}',if (n2>0 and n3<0,'{advice2}',if(n2<0 and n3>0,'{advice3}','{advice4}')))", "description": "", "templateType": "anything", "can_override": false}, "advice1": {"name": "advice1", "group": "Ungrouped variables", "definition": "\"<p>To rewrite $\\\\simplify{{n1}log({a})+{n2}log({b})+{n3}log({c})}$ as a single logarithm, we want to use the following rules:</p>\\n<ul>\\n<li>$n\\\\log(a)=\\\\log(a^n)$;</li>\\n<li>$ \\\\log(a)+\\\\log(b) = \\\\log(ab) $.</li>\\n</ul>\\n<p>Hence,</p>\\n<p>\\\\[ \\\\begin{split} \\\\simplify{{n1}log({a})+{n2}log({b})+{n3}log({c})} &amp;\\\\,= \\\\simplify[!collectNumbers]{log({a}^{n1})+log({b}^{n2})+log({c}^{n3})} \\\\\\\\ &amp;\\\\,= \\\\simplify{log({a^n1})+log({b^n2})+log({c^n3})} \\\\\\\\&amp;\\\\,=\\\\simplify{log({sol})}. \\\\end{split} \\\\]</p>\"", "description": "", "templateType": "long string", "can_override": false}, "advice2": {"name": "advice2", "group": "Ungrouped variables", "definition": "\"<p>To rewrite $\\\\simplify{{n1}log({a})+{n2}log({b})+{n3}log({c})}$ as a single logarithm, we want to use the following rules:</p>\\n<ul>\\n<li>$n\\\\log(a)=\\\\log(a^n)$;</li>\\n<li>$\\\\log(a)+\\\\log(b) = \\\\log(ab)$;</li>\\n<li>$ \\\\log(a)-\\\\log(b) = \\\\log\\\\left(\\\\frac{a}{b}\\\\right) $.</li>\\n</ul>\\n<p>Hence,</p>\\n<p>\\\\[ \\\\begin{split} \\\\simplify{{n1}log({a})+{n2}log({b})+{n3}log({c})} &amp;\\\\,= \\\\simplify[!collectNumbers]{log({a}^{n1})+log({b}^{n2})-log({c}^{abs(n3)})} \\\\\\\\ &amp;\\\\,=\\\\simplify{log({a^n1})+log({b^n2})-log({c^abs(n3)})} \\\\\\\\ &amp;\\\\,=\\\\simplify{log({a^n1}*{b^n2})-log({c^abs(n3)})}\\\\\\\\&nbsp; &amp;\\\\,=\\\\simplify[fractionNumbers]{log({sol})}. \\\\end{split} \\\\]</p>\"", "description": "", "templateType": "long string", "can_override": false}, "sol": {"name": "sol", "group": "Ungrouped variables", "definition": "if(n2>0 and n3>0,(a^n1)*(b^n2)*(c^n3),if(n2>0 and n3<0,((a^n1)*(b^n2))/(c^abs(n3)),if(n2<0 and n3>0,((a^n1)*(c^n3))/(b^abs(n2)),(a^n1)/((b^abs(n2))*(c^abs(n3))))))", "description": "", "templateType": "anything", "can_override": false}, "n2": {"name": "n2", "group": "Ungrouped variables", "definition": "random([-1,1])*random(2..3)", "description": "", "templateType": "anything", "can_override": false}, "n3": {"name": "n3", "group": "Ungrouped variables", "definition": "random([-1,1])*random(2..3)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(2..4 except[a,b])", "description": "", "templateType": "anything", "can_override": false}, "advice3": {"name": "advice3", "group": "Ungrouped variables", "definition": "\"<p>To rewrite $\\\\simplify{{n1}log({a})+{n2}log({b})+{n3}log({c})}$ as a single logarithm, we want to use the following rules:</p>\\n<ul>\\n<li>$n\\\\log(a)=\\\\log(a^n)$;</li>\\n<li>$\\\\log(a)+\\\\log(b) = \\\\log(ab)$;</li>\\n<li>$ \\\\log(a)-\\\\log(b) = \\\\log\\\\left(\\\\frac{a}{b}\\\\right) $.</li>\\n</ul>\\n<p>Hence,</p>\\n<p>\\\\[ \\\\begin{split} \\\\simplify{{n1}log({a})+{n2}log({b})+{n3}log({c})} &amp;\\\\,= \\\\simplify[!collectNumbers]{log({a}^{n1})-log({b}^{abs(n2)})+log({c}^{n3})} \\\\\\\\ &amp;\\\\,=\\\\simplify{log({a^n1})-log({b^abs(n2)})+log({c^n3})} \\\\\\\\ &amp;\\\\,=\\\\simplify{log({a^n1}*{c^n3})-log({b^abs(n2)})}\\\\\\\\&nbsp; &amp;\\\\,=\\\\simplify[fractionNumbers]{log({sol})}. \\\\end{split} \\\\]</p>\"", "description": "", "templateType": "long string", "can_override": false}, "advice4": {"name": "advice4", "group": "Ungrouped variables", "definition": "\"<p>To rewrite $\\\\simplify{{n1}log({a})+{n2}log({b})+{n3}log({c})}$ as a single logarithm, we want to use the following rules:</p>\\n<ul>\\n<li>$n\\\\log(a)=\\\\log(a^n)$;</li>\\n<li>$\\\\log(a)+\\\\log(b) = \\\\log(ab)$;</li>\\n<li>$ \\\\log(a)-\\\\log(b) = \\\\log\\\\left(\\\\frac{a}{b}\\\\right) $.</li>\\n</ul>\\n<p>Hence,</p>\\n<p>\\\\[ \\\\begin{split} \\\\simplify{{n1}log({a})+{n2}log({b})+{n3}log({c})} &amp;\\\\,= \\\\simplify[!collectNumbers]{log({a}^{n1})-log({b}^{abs(n2)})-log({c}^{abs(n3)})} \\\\\\\\ &amp;\\\\,=\\\\simplify{log({a^n1})-log({b^abs(n2)})-log({c^abs(n3)})} \\\\\\\\ &amp;\\\\,=\\\\simplify{log({a^n1})-log({b^abs(n2)}*{c^abs(n3)})}\\\\\\\\&nbsp; &amp;\\\\,=\\\\simplify[fractionNumbers]{log({sol})}. \\\\end{split} \\\\]</p>\"", "description": "", "templateType": "long string", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["n1", "n2", "n3", "a", "b", "c", "advice", "advice1", "advice2", "advice3", "advice4", "sol"], "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": "log({sol})", "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/"}]}