// Numbas version: finer_feedback_settings
{"name": "Logaritmer - forenkle uttrykk 3", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Logaritmer - forenkle uttrykk 3", "tags": [], "metadata": {"description": "<p>Simlify (expand) logarithmic expressions&nbsp;</p>", "licence": "None specified"}, "statement": "<p>Forenkle uttrykket.</p>", "advice": "<p>Her bruker vi logaritmesetningene:</p>\n<ol>\n<li>$\\ln(ab)=\\ln(a)+\\ln(b)$</li>\n<li>$\\ln\\left(\\dfrac{a}{b}\\right)=\\ln(a)-\\ln(b)$</li>\n<li>$\\ln\\left(a^{b}\\right)=b\\ln(a)$</li>\n</ol>\n<p>I tillegg bruker vi sammenhengen $\\ln(e)=1$.</p>\n<p>Vi skal forenkle uttrykket</p>\n<p>\\[\\ln\\left(\\var{b1}a^{\\var{c[0]}}\\right)-\\ln\\left(\\dfrac{b^{\\var{c[1]}}}{a^{\\var{c[2]}}}\\right)+\\ln\\left(\\dfrac{e^{\\var{c[3]}}}{\\var{b2}b^{\\var{c[4]}}}\\right)\\]</p>\n<p>Her kan det v&aelig;re en fordel &aring; forenkle hvert ledd for seg f&oslash;rst:</p>\n<ul>\n<li>$\\ln\\left(\\var{b1}a^{\\var{c[0]}}\\right)=\\ln(\\var{b1})+\\ln(a^{\\var{c[0]}})=\\ln(2^{\\var{ex[0]}})+\\var{c[0]}\\ln(a)=\\var{ex[0]}\\ln(2)+\\var{c[0]}\\ln(a)$</li>\n<li>$\\ln\\left(\\dfrac{b^{\\var{c[1]}}}{a^{\\var{c[2]}}}\\right)=\\ln(b^{\\var{c[1]}})-\\ln(a^{\\var{c[2]}})=\\var{c[1]}\\ln(b)-\\var{c[2]}\\ln(a)$</li>\n<li>$\\ln\\left(\\dfrac{e^{\\var{c[3]}}}{\\var{b2}b^{\\var{c[4]}}}\\right)=\\ln(e^{\\var{c[3]}})-\\ln(\\var{b2}b^{\\var{c[4]}})=\\var{c[3]}\\ln(e)-\\left(\\ln(\\var{b2})+\\ln(b^{\\var{c[4]}})\\right)=\\var{c[3]}-\\ln(2^{\\var{ex[1]}})-\\ln(b^{\\var{c[4]}})=\\var{c[3]}-\\var{ex[1]}\\ln(2)-\\var{c[4]}\\ln(b)$</li>\n</ul>\n<p>S&aring; setter vi sammen:</p>\n<p>$\\ln\\left(\\var{b1}a^{\\var{c[0]}}\\right)-\\ln\\left(\\dfrac{b^{\\var{c[1]}}}{a^{\\var{c[2]}}}\\right)+\\ln\\left(\\dfrac{e^{\\var{c[3]}}}{\\var{b2}b^{\\var{c[4]}}}\\right)\\\\<br/>=\\var{ex[0]}\\ln(2)+\\var{c[0]}\\ln(a)-\\left(\\var{c[1]}\\ln(b)-\\var{c[2]}\\ln(a)\\right)+\\var{c[3]}-\\var{ex[1]}\\ln(2)-\\var{c[4]}\\ln(b)\\\\<br/>=\\var{c[0]}\\ln(a)+\\var{c[2]}\\ln(a)-\\var{c[1]}\\ln(b)-\\var{c[4]}\\ln(b)+\\var{ex[0]}\\ln(2)-\\var{ex[1]}\\ln(2)+\\var{c[3]}\\\\<br/>=\\underline{\\underline{\\var{ka}\\ln(a)-\\var{kb}\\ln(b)+\\var{k2}\\ln(2)+\\var{c[3]}}}$</p>\n<p></p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"c": {"name": "c", "group": "Ungrouped variables", "definition": "repeat(random(2..7),6)", "description": "", "templateType": "anything", "can_override": false}, "ex": {"name": "ex", "group": "Ungrouped variables", "definition": "repeat(random(1..5),2)", 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{"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\ln\\left(\\var{b1}a^{\\var{c[0]}}\\right)-\\ln\\left(\\dfrac{b^{\\var{c[1]}}}{a^{\\var{c[2]}}}\\right)+\\ln\\left(\\dfrac{e^{\\var{c[3]}}}{\\var{b2}b^{\\var{c[4]}}}\\right)$=[[0]]$\\ln(a)-$[[1]]$\\ln(b)+$[[2]]$\\ln(2)+$[[3]]</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", 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