// Numbas version: finer_feedback_settings
{"name": "Logarithms (laws)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"rulesets": {}, "variable_groups": [], "tags": [], "name": "Logarithms (laws)", "functions": {}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Use laws for addition and subtraction of logarithms to simplify a given logarithmic expression to an arbitrary base.</p>"}, "ungrouped_variables": ["x1", "y1", "y2", "y3"], "parts": [{"variableReplacementStrategy": "originalfirst", "steps": [{"variableReplacementStrategy": "originalfirst", "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "showCorrectAnswer": true, "showFeedbackIcon": true, "type": "information", "unitTests": [], "variableReplacements": [], "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>", "scripts": {}, "marks": 0}], "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "sortAnswers": false, "variableReplacements": [], "gaps": [{"notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "unitTests": [], "variableReplacements": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "minValue": "x1[1]*x1[0]", "scripts": {}, "marks": "2", "extendBaseMarkingAlgorithm": true, "mustBeReduced": false, "customMarkingAlgorithm": "", "allowFractions": false, "type": "numberentry", "correctAnswerFraction": false, "maxValue": "x1[1]*x1[0]"}], "marks": 0, "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "scripts": {}, "stepsPenalty": 0, "type": "gapfill", "prompt": "<p>$\\log_a(\\var{x1[1]})+ \\log_a(\\var{x1[0]})=\\log_a($ [[0]]$)$</p>"}, {"variableReplacementStrategy": "originalfirst", "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "sortAnswers": false, "variableReplacements": [], "gaps": [{"notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "unitTests": [], "variableReplacements": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "minValue": "y1", "scripts": {}, "marks": "2", "extendBaseMarkingAlgorithm": true, "mustBeReduced": false, "customMarkingAlgorithm": "", "allowFractions": false, "type": "numberentry", "correctAnswerFraction": false, "maxValue": "y1"}], "marks": 0, "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "scripts": {}, "type": "gapfill", "prompt": "<p>$\\log_a(\\var{(x1[4])*y1})-\\log_a(\\var{x1[4]})=\\log_a($ [[0]]$)$</p>"}, {"variableReplacementStrategy": "originalfirst", "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "sortAnswers": false, "variableReplacements": [], "gaps": [{"notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "unitTests": [], "variableReplacements": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "minValue": "y2*y3", "scripts": {}, "marks": "2", "extendBaseMarkingAlgorithm": true, "mustBeReduced": false, "customMarkingAlgorithm": "", "allowFractions": false, "type": "numberentry", "correctAnswerFraction": false, "maxValue": "y2*y3"}], "marks": 0, "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "scripts": {}, "type": "gapfill", "prompt": "<p>$\\log_a(\\var{y2})+\\log_a(\\var{(x1[4])*y3})-\\log_a(\\var{x1[4]})=\\log_a($[[0]]$)$</p>"}], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"y3": {"description": "", "templateType": "anything", "definition": "random(2..6)", "name": "y3", "group": "Ungrouped variables"}, "y2": {"description": "<p>random(2..6)</p>", "templateType": "anything", "definition": "random(2..7)", "name": "y2", "group": "Ungrouped variables"}, "x1": {"description": "", "templateType": "anything", "definition": "repeat(random(2..20),8)", "name": "x1", "group": "Ungrouped variables"}, "y1": {"description": "", "templateType": "anything", "definition": "random(2..6)", "name": "y1", "group": "Ungrouped variables"}}, "extensions": [], "type": "question", "preamble": {"js": "", "css": ""}, "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>", "statement": "<p>Simplify the expressions to fill in the gaps.</p>", "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/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}]}], "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/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}