// Numbas version: exam_results_page_options
{"name": "Simplify logarithms", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variablesTest": {"condition": "", "maxRuns": 100}, "variables": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "name": "s1", "description": ""}, "d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s4*random(2..9)", "name": "d", "description": ""}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "name": "c", "description": ""}, "f": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround((a2-1)/b2,0)", "name": "f", "description": ""}, "s4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "name": "s4", "description": ""}, "b1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..15)", "name": "b1", "description": ""}, "a2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "1+b2*random(2..5)", "name": "a2", "description": ""}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(2..9)", "name": "a1", "description": ""}, "b2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..9)", "name": "b2", "description": ""}}, "ungrouped_variables": ["c", "d", "f", "s1", "s4", "a1", "a2", "b1", "b2"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "name": "Simplify logarithms", "showQuestionGroupNames": false, "variable_groups": [], "functions": {}, "parts": [{"scripts": {}, "gaps": [{"showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "{a1}", "minValue": "{a1}", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "{b1}", "minValue": "{b1}", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}], "type": "gapfill", "prompt": "\n      <p>Express the following in terms of $\\log_a(x)$ and $\\log_a(y)$</p>\n        <p>\\[\\log_a(x^{\\var{a1}}y^{\\var{b1}})=\\alpha\\log_a(x)+\\beta\\log_a(y)\\]</p>\n        <p>$\\alpha=\\;\\;$[[0]], $\\beta=\\;\\;$[[1]]</p>\n      ", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"answer": "((x ^ {f}) * (({c} * x) + {d}))", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "answersimplification": "std", "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\n      <p>\\[\\simplify[std]{{a2}/{b2}}\\log_a(x)+\\log_a(\\simplify{{c}*x+{d}})-\\log_a(\\simplify{x^(1/{b2})})=\\log_a(q(x))\\]</p>\n        <p>$q(x)=\\;\\;$[[0]]</p>\n      ", "showCorrectAnswer": true, "marks": 0}], "statement": "<p>Answer the following questions on logarithms.</p>", "tags": ["checked2015", "log laws", "logarithm laws", "logarithmic expressions", "logarithms", "logs", "MAS1601", "mas1601", "rules for logarithms", "simplifying logarithms"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "<p><strong>2/06/2012:</strong></p>\n            <p>Added tags.</p>\n            <p>Changed statement to make question clearer.</p>\n            <p><strong>19/07/2012:</strong></p>\n            <p>Added description.</p>\n            <p><strong>25/07/2012:</strong></p>\n            <p>Added tags.</p>\n            <p>Question appears to be working correctly.</p>\n            <p><strong>17/08/2012:</strong></p>\n            <p>Made copy to include in Simplify Algebraic Expressions exam.</p>", "licence": "Creative Commons Attribution 4.0 International", "description": "\n            \t\t<p>Express&nbsp;$\\log_a(x^{c}y^{d})$&nbsp;in terms of $\\log_a(x)$ and $\\log_a(y)$. Find $q(x)$ such that $\\frac{f}{g}\\log_a(x)+\\log_a(rx+s)-\\log_a(x^{1/t})=\\log_a(q(x))$</p>\n            \t\t    <p>&nbsp;</p>\n            \t\t"}, "advice": "<p>The rules for combining logs are</p>\n  <p>\\[\\begin{eqnarray*} \\log_a(bc)&amp;=&amp;\\log_a(b)+\\log_a(c)\\\\ \\\\ \\log_a\\left(\\frac{b}{c}\\right)&amp;=&amp;\\log_a(b)-\\log_a(c)\\\\ \\\\ \\log_a(b^r)&amp;=&amp;r\\log_a(b) \\end{eqnarray*} \\]</p>\n  <p>a)<br/>Using these rules gives:<br/>\\[ \\begin{eqnarray*} \\log_a(x^{\\var{a1}}y^{\\var{b1}})&amp;=&amp;\\log_a(x^{\\var{a1}})+\\log_a(y^{\\var{b1}})\\\\ &amp;=&amp;\\var{a1}\\log_a(x)+\\var{b1}\\log_a(y) \\end{eqnarray*} \\]</p>\n  <p>b)<br/>\\[\\begin{eqnarray*} \\simplify[std]{{a2}/{b2}}\\log_a(x)+\\log_a(\\simplify{{c}*x+{d}})-\\log_a(\\simplify{x^(1/{b2})})&amp;=&amp;\\log_a(x^\\frac{\\var{a2}}{\\var{b2}})+\\log_a(\\simplify{{c}*x+{d}})-\\log_a(\\simplify{x^(1/{b2})})\\\\ \\\\ &amp;=&amp;\\log_a\\left(\\simplify[std]{(x^({a2}/{b2})*({c}x+{d}))/(x^(1/{b2}))}\\right)\\\\ &amp;=&amp;\\log_a\\left(\\simplify{x^{f}*({c}x+{d})}\\right) \\end{eqnarray*} \\]</p>", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}