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{"name": "NUMBAS-Logarithms (laws)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"parts": [{"steps": [{"customMarkingAlgorithm": "", "showCorrectAnswer": true, "marks": 0, "variableReplacementStrategy": "originalfirst", "extendBaseMarkingAlgorithm": true, "scripts": {}, "variableReplacements": [], "type": "information", "unitTests": [], "prompt": "<p>When adding and subtracting logarithms we can simplify the expressions using some logarithm laws. These laws are</p>\n<p>\\[\\begin{align}<br/>\\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>", "showFeedbackIcon": true}], "prompt": "<p>$\\log_a(\\var{x1[1]})+ \\log_a(\\var{x1[0]})=\\log_a($ [[0]]$)$</p>", "showCorrectAnswer": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "stepsPenalty": 0, "sortAnswers": false, "marks": 0, "gaps": [{"maxValue": "x1[1]*x1[0]", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "unitTests": [], "showFeedbackIcon": true, "mustBeReducedPC": 0, "correctAnswerStyle": "plain", "marks": "2", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "correctAnswerFraction": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "minValue": "x1[1]*x1[0]"}], "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "scripts": {}, "type": "gapfill", "unitTests": []}, {"prompt": "<p>$\\log_a(\\var{(x1[4])*y1})-\\log_a(\\var{x1[4]})=\\log_a($ [[0]]$)$</p>", "showCorrectAnswer": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "sortAnswers": false, "marks": 0, "gaps": [{"maxValue": "y1", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "unitTests": [], "showFeedbackIcon": true, "mustBeReducedPC": 0, "correctAnswerStyle": "plain", "marks": "2", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "correctAnswerFraction": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "minValue": "y1"}], "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "scripts": {}, "type": "gapfill", "unitTests": []}, {"prompt": "<p>$\\log_a(\\var{y2})+\\log_a(\\var{(x1[4])*y3})-\\log_a(\\var{x1[4]})=\\log_a($[[0]]$)$</p>", "showCorrectAnswer": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "sortAnswers": false, "marks": 0, "gaps": [{"maxValue": "y2*y3", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "unitTests": [], "showFeedbackIcon": true, "mustBeReducedPC": 0, "correctAnswerStyle": "plain", "marks": "2", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "correctAnswerFraction": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "minValue": "y2*y3"}], "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "scripts": {}, "type": "gapfill", "unitTests": []}], "name": "NUMBAS-Logarithms (laws)", "variablesTest": {"maxRuns": 100, "condition": ""}, "rulesets": {}, "tags": [], "variable_groups": [], "extensions": [], "functions": {}, "metadata": {"description": "<p>Use laws for addition and subtraction of logarithms to simplify a given logarithmic expression to an arbitrary base.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Simplify the expressions to fill in the gaps.</p>", "variables": {"x1": {"group": "Ungrouped variables", "templateType": "anything", "name": "x1", "definition": "repeat(random(2..20),8)", "description": ""}, "y2": {"group": "Ungrouped variables", "templateType": "anything", "name": "y2", "definition": "random(2..7)", "description": "<p>random(2..6)</p>"}, "y1": {"group": "Ungrouped variables", "templateType": "anything", "name": "y1", "definition": "random(2..6)", "description": ""}, "y3": {"group": "Ungrouped variables", "templateType": "anything", "name": "y3", "definition": "random(2..6)", "description": ""}}, "advice": "<h4>a)</h4>\n<p>We need to use the rule</p>\n<p>\\[\\log_a(x)+\\log_a(y)=\\log_a(xy)\\text{.}\\]</p>\n<p>Substituting in our values for $x$ and $y$ gives</p>\n<p>\\[\\begin{align}<br/>\\log_a(\\var{x1[1]})+\\log_a(\\var{x1[0]})&amp;=\\log_a(\\var{x1[1]}\\times \\var{x1[0]})\\\\<br/>&amp;=\\log_a(\\var{x1[1]*x1[0]})\\text{.}<br/>\\end{align}\\]</p>\n<p><em></em></p>\n<h4>b)</h4>\n<p>We need to use the rule</p>\n<p>\\[\\log_a(x)-\\log_a(y)=\\log_a\\left(\\frac{x}{y}\\right)\\text{.}\\]</p>\n<p>Substituting in our values for $x$ and $y$ gives</p>\n<p>\\[\\begin{align}<br/>\\log_a(\\var{x1[4]*y1})-\\log_a(\\var{x1[4]})&amp;=\\log_a(\\var{x1[4]*y1}\\div \\var{x1[4]})\\\\<br/>&amp;=\\log_a(\\var{y1})\\text{.}<br/>\\end{align}\\]</p>\n<h4><em></em><em></em></h4>", "ungrouped_variables": ["x1", "y1", "y2", "y3"], "preamble": {"js": "", "css": ""}, "type": "question", "contributors": [{"name": "Katy Dobson", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/854/"}, {"name": "Aiden McCall", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1592/"}, {"name": "Hannah Aldous", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1594/"}]}]}], "contributors": [{"name": "Katy Dobson", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/854/"}, {"name": "Aiden McCall", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1592/"}, {"name": "Hannah Aldous", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1594/"}]}