// Numbas version: exam_results_page_options
{"name": "Addition and Subtraction of Logarithms", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variables": {"y1": {"definition": "random(2..6)", "group": "Ungrouped variables", "description": "", "name": "y1", "templateType": "anything"}, "x1": {"definition": "repeat(random(2..20),8)", "group": "Ungrouped variables", "description": "", "name": "x1", "templateType": "anything"}}, "parts": [{"showCorrectAnswer": true, "showFeedbackIcon": true, "marks": 0, "type": "gapfill", "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "prompt": "<p>$\\log_a(\\var{x1[1]})+ \\log_a(\\var{x1[0]})=\\log_a($ [[0]]$)$</p>", "gaps": [{"variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "type": "numberentry", "scripts": {}, "variableReplacements": [], "mustBeReducedPC": 0, "showCorrectAnswer": true, "allowFractions": false, "marks": "1", "correctAnswerFraction": false, "correctAnswerStyle": "plain", "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "maxValue": "x1[1]*x1[0]", "minValue": "x1[1]*x1[0]"}]}, {"showCorrectAnswer": true, "showFeedbackIcon": true, "marks": 0, "type": "gapfill", "scripts": {}, "variableReplacements": [], "stepsPenalty": 0, "variableReplacementStrategy": "originalfirst", "prompt": "<p>$\\log_a(\\var{(x1[4])*y1})-\\log_a(\\var{x1[4]})=\\log_a($ [[0]]$)$</p>", "steps": [{"showCorrectAnswer": true, "showFeedbackIcon": true, "marks": 0, "type": "information", "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "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>"}], "gaps": [{"variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "type": "numberentry", "scripts": {}, "variableReplacements": [], "mustBeReducedPC": 0, "showCorrectAnswer": true, "allowFractions": false, "marks": "1", "correctAnswerFraction": false, "correctAnswerStyle": "plain", "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "maxValue": "y1", "minValue": "y1"}]}], "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>", "rulesets": {}, "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"}, "extensions": [], "tags": [], "ungrouped_variables": ["x1", "y1"], "functions": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "preamble": {"css": "", "js": ""}, "variable_groups": [], "name": "Addition and Subtraction of Logarithms", "statement": "<p>Trekk sammen uttrykkene.&nbsp;</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>", "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/"}]}