// Numbas version: exam_results_page_options
{"name": "Logarithms: Rewriting log expressions as a single log", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"rulesets": {}, "variable_groups": [], "tags": [], "parts": [{"variableReplacementStrategy": "originalfirst", "steps": [{"variableReplacementStrategy": "originalfirst", "customName": "", "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "scripts": {}, "marks": 0, "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "useCustomName": false, "unitTests": [], "type": "information", "prompt": "<p>The numbers multiplying the logs (sign and all!) can be put inside the log by using $\\log(a^b)=b\\log(a)$:</p>\n<p>\\[ \\simplify[!collectNumbers]{{c1}log({a1},{b})+{c2}log({a2},{b})+{c3}log({a3},{b})=log({a1}^{c1},{b})+log({a2}^{c2},{b})+log({a3}^{c3},{b})}.\\]</p>\n<p>Then we can&nbsp;add the logs using $\\log(a)+\\log(b)=\\log(ab)$:</p>\n<p>\\[ \\begin{align}\\simplify[!collectNumbers]{log({a1}^{c1},{b})+log({a2}^{c2},{b})+log({a3}^{c3},{b})}&amp;=\\simplify[!collectNumbers]{log({a1}^{c1}{a2}^{c2},{b})+log({a3}^{c3},{b})}\\\\&amp;=\\simplify[!collectNumbers]{log({a3}^{c3}*{a1}^{c1}*{a2}^{c2},{b})}.\\end{align}\\]</p>\n<p><span data-jme-visible=\"c1&lt;0 or c2&lt;0 or c3&lt;0\">Note, if we wanted to, we could write our answer without negative indices (recall $b^{-a}=\\frac{1}{b^a}$):</span></p>\n<p><span data-jme-visible=\"c1&lt;0 or c2&lt;0 or c3&lt;0\">\\[\\log_\\var{b}\\left(\\simplify[!collectNumbers,zeroPower,unitFactor,unitDenominator,unitPower]{{a1}^{(if(c1&gt;0,c1,0))}{a2}^{(if(c2&gt;0,c2,0))}{a3}^{(if(c3&gt;0,c3,0))}/({a1}^{(if(c1&lt;0,-c1,0))}{a2}^{(if(c2&lt;0,-c2,0))}{a3}^{(if(c3&lt;0,-c3,0))})}\\right)\\]</span></p>\n<p><span data-jme-visible=\"c1&lt;0 or c2&lt;0 or c3&lt;0\"></span></p>\n<p><span data-jme-visible=\"c1&lt;0 or c2&lt;0 or c3&lt;0\"><strong>Note</strong> we could approach this question slightly differently and made use of&nbsp;$\\log(a)-\\log(b)=\\log\\left(\\frac{a}{b}\\right)$ as well to deal with the logs with negatives out the front but it isn't necessary.&nbsp;</span></p>\n<p></p>"}], "customName": "", "showCorrectAnswer": true, "showFeedbackIcon": true, "sortAnswers": false, "variableReplacements": [], "scripts": {}, "marks": 0, "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "useCustomName": false, "stepsPenalty": "1", "gaps": [{"variableReplacementStrategy": "originalfirst", "unitTests": [], "showFeedbackIcon": true, "vsetRangePoints": 5, "variableReplacements": [], "useCustomName": false, "marks": 1, "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "vsetRange": [0, 1], "showCorrectAnswer": true, "type": "jme", "showPreview": true, "valuegenerators": [], "checkVariableNames": false, "failureRate": 1, "answerSimplification": "!collectNumbers", "customName": "", "scripts": {}, "checkingAccuracy": 0.001, "customMarkingAlgorithm": "", "answer": "{a1}^{c1}*{a2}^{c2}*{a3}^{c3}"}], "unitTests": [], "type": "gapfill", "prompt": "<p>$\\displaystyle \\simplify{{c1}log({a1},{b})+{c2}log({a2},{b})+{c3}log({a3},{b})}=\\log_\\var{b}{\\Large(}$ [[0]]$\\Large)$</p>"}, {"variableReplacementStrategy": "originalfirst", "steps": [{"variableReplacementStrategy": "originalfirst", "customName": "", "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "scripts": {}, "marks": 0, "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "useCustomName": false, "unitTests": [], "type": "information", "prompt": "<p>Recall that $\\ln$ denotes&nbsp;$\\log_e$.</p>\n<p>We use the standard log and index laws to manipulate this expression into a single log term:</p>\n<p style=\"text-align: center;\">$\\begin{align}\\simplify{1/{a1}ln(d)+{a2}ln(d^{a3})+{n}ln(d)}&amp;=\\simplify{ln(d^(1/{a1})) + ln((d^{a3})^{a2})+ln(d^{n})}\\\\ &amp;=\\simplify{ln(d^(1/{a1})) + ln(d^{a3*a2})+ln(d^{n})}\\\\&amp;=\\simplify[!collectNumbers]{ln(d^(1/{a1})d^{a3*a2}d^{n})}\\\\&amp;=\\simplify[!collectNumbers]{ln(d^(1/{a1}+{a3*a2}+{n}))}\\\\&amp;=\\simplify[fractionNumbers,simplifyFractions,unitDenominator]{ln(d^{1/a1+a3*a2+n})}\\end{align}$</p>"}], "customName": "", "showCorrectAnswer": true, "showFeedbackIcon": true, "sortAnswers": false, "variableReplacements": [], "scripts": {}, "marks": 0, "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "useCustomName": false, "stepsPenalty": "1", "gaps": [{"variableReplacementStrategy": "originalfirst", "unitTests": [], "showFeedbackIcon": true, "vsetRangePoints": 5, "variableReplacements": [], "useCustomName": false, "marks": 1, "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "vsetRange": [0, 1], "showCorrectAnswer": true, "type": "jme", "showPreview": true, "valuegenerators": [{"name": "d", "value": ""}], "checkVariableNames": false, "failureRate": 1, "answerSimplification": "collectNumbers, fractionNumbers, simplifyFractions", "customName": "", "scripts": {}, "checkingAccuracy": 0.001, "customMarkingAlgorithm": "", "answer": "d^({1/{a1}+{a3}*{a2}+{n}})"}], "unitTests": [], "type": "gapfill", "prompt": "<p>$\\displaystyle &nbsp;\\simplify{1/{a1}ln(d)+{a2}ln(d^{a3})+{n}ln(d)}=\\ln\\Large($ [[0]] $\\Large)$&nbsp;</p>"}], "functions": {}, "name": "Logarithms: Rewriting log expressions as a single log", "ungrouped_variables": ["c", "c1", "c2", "c3", "a", "a1", "a2", "a3", "b", "n"], "preamble": {"js": "", "css": ""}, "variablesTest": {"maxRuns": 100, "condition": ""}, "metadata": {"licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International", "description": "<p>Testing whether students know how to use multiple log laws.</p>"}, "extensions": [], "statement": "<p>Rewrite the following expressions involving logs as a single logarithm</p>", "advice": "<p>This question requires you to know the log laws and the index laws.</p>\n<p></p>", "variables": {"c": {"description": "", "definition": "shuffle(-7..7 except [1,0])[0..3]", "group": "Ungrouped variables", "name": "c", "templateType": "anything"}, "c2": {"description": "", "definition": "c[1]", "group": "Ungrouped variables", "name": "c2", "templateType": "anything"}, "c3": {"description": "", "definition": "c[2]", "group": "Ungrouped variables", "name": "c3", "templateType": "anything"}, "a2": {"description": "", "definition": "a[1]", "group": "Ungrouped variables", "name": "a2", "templateType": "anything"}, "a3": {"description": "", "definition": "a[2]", "group": "Ungrouped variables", "name": "a3", "templateType": "anything"}, "c1": {"description": "", "definition": "c[0]", "group": "Ungrouped variables", "name": "c1", "templateType": "anything"}, "n": {"description": "", "definition": "random(-7..-1)", "group": "Ungrouped variables", "name": "n", "templateType": "anything"}, "b": {"description": "", "definition": "random(e,2,10,3,5,6)", "group": "Ungrouped variables", "name": "b", "templateType": "anything"}, "a": {"description": "", "definition": "shuffle(2..7)[0..3]", "group": "Ungrouped variables", "name": "a", "templateType": "anything"}, "a1": {"description": "", "definition": "a[0]", "group": "Ungrouped variables", "name": "a1", "templateType": "anything"}}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}]}], "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}