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{"name": "Using Laws for Addition and Subtraction of Logarithms [L6 Randomised]", "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": {"x1": {"definition": "repeat(random(2..20),8)", "name": "x1", "description": "", "templateType": "anything", "group": "Ungrouped variables"}, "y1": {"definition": "random(2..6)", "name": "y1", "description": "", "templateType": "anything", "group": "Ungrouped variables"}}, "preamble": {"css": "", "js": ""}, "extensions": [], "rulesets": {}, "variable_groups": [], "parts": [{"showCorrectAnswer": true, "steps": [{"customMarkingAlgorithm": "", "showFeedbackIcon": true, "scripts": {}, "showCorrectAnswer": true, "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>", "variableReplacementStrategy": "originalfirst", "unitTests": [], "marks": 0, "type": "information", "extendBaseMarkingAlgorithm": true, "variableReplacements": []}], "type": "gapfill", "marks": 0, "gaps": [{"maxValue": "x1[1]*x1[0]", "showCorrectAnswer": true, "correctAnswerStyle": "plain", "allowFractions": false, "mustBeReducedPC": 0, "type": "numberentry", "marks": "2", "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "customMarkingAlgorithm": "", "showFeedbackIcon": true, "scripts": {}, "minValue": "x1[1]*x1[0]", "notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "correctAnswerFraction": false, "variableReplacementStrategy": "originalfirst", "unitTests": []}], "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "customMarkingAlgorithm": "", "showFeedbackIcon": true, "scripts": {}, "prompt": "<p>$\\log_a(\\var{x1[1]})+ \\log_a(\\var{x1[0]})=\\log_a($ [[0]]$)$</p>", "sortAnswers": false, "variableReplacementStrategy": "originalfirst", "unitTests": [], "stepsPenalty": 0}, {"showCorrectAnswer": true, "steps": [{"customMarkingAlgorithm": "", "showFeedbackIcon": true, "scripts": {}, "showCorrectAnswer": true, "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>", "variableReplacementStrategy": "originalfirst", "unitTests": [], "marks": 0, "type": "information", "extendBaseMarkingAlgorithm": true, "variableReplacements": []}], "type": "gapfill", "marks": 0, "gaps": [{"maxValue": "y1", "showCorrectAnswer": true, "correctAnswerStyle": "plain", "allowFractions": false, "mustBeReducedPC": 0, "type": "numberentry", "marks": "2", "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "customMarkingAlgorithm": "", "showFeedbackIcon": true, "scripts": {}, "minValue": "y1", "notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "correctAnswerFraction": false, "variableReplacementStrategy": "originalfirst", "unitTests": []}], "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "customMarkingAlgorithm": "", "showFeedbackIcon": true, "scripts": {}, "prompt": "<p>$\\log_a(\\var{(x1[4])*y1})-\\log_a(\\var{x1[4]})=\\log_a($ [[0]]$)$</p>", "sortAnswers": false, "variableReplacementStrategy": "originalfirst", "unitTests": [], "stepsPenalty": 0}], "tags": ["L6"], "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>", "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"}, "ungrouped_variables": ["x1", "y1"], "statement": "<p>Simplify the expressions to fill in the gaps.</p>", "name": "Using Laws for Addition and Subtraction of Logarithms [L6 Randomised]", "functions": {}, "type": "question", "contributors": [{"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": "Matthew James Sykes", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2582/"}]}]}], "contributors": [{"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": "Matthew James Sykes", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2582/"}]}