// Numbas version: exam_results_page_options
{"name": "Ida's copy of Simplify logarithms", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"extensions": [], "variable_groups": [], "preamble": {"css": "", "js": ""}, "functions": {}, "ungrouped_variables": ["c", "d", "f", "s1", "s4", "a1", "a2", "b1", "b2"], "name": "Ida's copy of Simplify logarithms", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "statement": "<p>Finn $\\alpha$ og $\\beta$.</p>\n<p>Vi har disse reglene, som gjelder for alle tall $b$ og alle positive tall $x$ og $y$:</p>\n<p>\\[\\begin{align}\\log_a(x^b)&amp;=b\\log(x)\\\\ \\log_a(x)+\\log_a(y)&amp;=\\log_a(xy)\\text{,}\\\\<br/>\\log_a(x)-\\log_a(y)&amp;=\\log_a\\left(\\frac{x}{y}\\right)\\text{.}<br/>\\end{align}\\]</p>", "metadata": {"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"}, "variablesTest": {"maxRuns": 100, "condition": ""}, "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>", "parts": [{"variableReplacements": [], "variableReplacementStrategy": "originalfirst", "marks": 0, "showCorrectAnswer": true, "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      ", "type": "gapfill", "scripts": {}, "showFeedbackIcon": true, "gaps": [{"allowFractions": false, "correctAnswerFraction": false, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "maxValue": "{a1}", "showCorrectAnswer": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "marks": 0.5, "minValue": "{a1}", "mustBeReducedPC": 0, "scripts": {}, "showFeedbackIcon": true, "correctAnswerStyle": "plain", "mustBeReduced": false}, {"allowFractions": false, "correctAnswerFraction": false, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "maxValue": "{b1}", "showCorrectAnswer": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "marks": 0.5, "minValue": "{b1}", "mustBeReducedPC": 0, "scripts": {}, "showFeedbackIcon": true, "correctAnswerStyle": "plain", "mustBeReduced": false}]}], "variables": {"s4": {"definition": "random(1,-1)", "name": "s4", "description": "", "templateType": "anything", "group": "Ungrouped variables"}, "a2": {"definition": "1+b2*random(2..5)", "name": "a2", "description": "", "templateType": "anything", "group": "Ungrouped variables"}, "b2": {"definition": "random(2..9)", "name": "b2", "description": "", "templateType": "anything", "group": "Ungrouped variables"}, "a1": {"definition": "s1*random(2..9)", "name": "a1", "description": "", "templateType": "anything", "group": "Ungrouped variables"}, "c": {"definition": "random(1..9)", "name": "c", "description": "", "templateType": "anything", "group": "Ungrouped variables"}, "s1": {"definition": "random(1,-1)", "name": "s1", "description": "", "templateType": "anything", "group": "Ungrouped variables"}, "d": {"definition": "s4*random(2..9)", "name": "d", "description": "", "templateType": "anything", "group": "Ungrouped variables"}, "f": {"definition": "precround((a2-1)/b2,0)", "name": "f", "description": "", "templateType": "anything", "group": "Ungrouped variables"}, "b1": {"definition": "random(2..15)", "name": "b1", "description": "", "templateType": "anything", "group": "Ungrouped variables"}}, "tags": [], "type": "question", "contributors": [{"name": "Ida Landg\u00e4rds", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2336/"}]}]}], "contributors": [{"name": "Ida Landg\u00e4rds", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2336/"}]}