// Numbas version: exam_results_page_options
{"name": "Ugur's copy of Logarithms1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Ugur's copy of Logarithms1", "tags": [], "metadata": {"description": "\n    \t\t<p>Given a sum of logs, all numbers are integers,</p>\n    \t\t                <p>$\\log_b(a_1)+\\alpha\\log_b(a_2)+\\beta\\log_b(a_3)$ write as $\\log_b(a)$ for some fraction $a$.</p>\n    \t\t                <p>Also calculate to 3 decimal places $\\log_b(a)$.&nbsp;</p>\n    \t\t", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Answer the following question on logarithms.</p>", "advice": "\n  <p>The rules for combining logs are</p>\n  <p>\\[\\begin{eqnarray*}&amp;1.&amp; &nbsp;\\log_b(ac)&amp;=&amp;\\log_b(a)+\\log_b(c)\\\\ \\\\ &amp;2.&amp; &nbsp;\\log_b\\left(\\frac{a}{c}\\right)&amp;=&amp;\\log_b(a)-\\log_b(c)\\\\ \\\\ &amp;3.&amp; &nbsp;\\log_b(a^r)&amp;=&amp;r\\log_b(a) \\end{eqnarray*} \\]</p>\n  <p>We see that:</p>\n  <p>\\[\\begin{eqnarray*}\\log_{\\var{b}}(\\var{a_1})-\\var{r_1}\\log_{\\var{b}}(\\var{a_2})+\\var{r_2}\\log_{\\var{b}}(\\var{a_3})&amp;=&amp;\\log_{\\var{b}}(\\var{a_1})-\\log_{\\var{b}}(\\var{a_2}^{\\var{r_1}})+\\log_{\\var{b}}(\\var{a_3}^{\\var{r_2}})\\mbox{ using 3.}\\\\&amp;=&amp;\\log_{\\var{b}}(\\var{a_1})-\\log_{\\var{b}}(\\var{a_2^r_1})+\\log_{\\var{b}}(\\var{a_3^r_2})\\\\&amp;=&amp;\\log_{\\var{b}}(\\var{a_1}\\times \\var{a_3^r_2})-\\log_{\\var{b}}(\\var{a_2^r_1}) \\mbox{ using 1.}\\\\&amp;=&amp;\\log_{\\var{b}}\\left(\\frac{\\var{a_1}\\times \\var{a_3^r_2}}{\\var{a_2^r_1}}\\right) \\mbox{ using 2.}\\\\&amp;=&amp;\\log_{\\var{b}}\\left(\\frac{\\var{a_1*a_3^r_2}}{\\var{a_2^r_1}}\\right)\\\\&amp;=&amp;\\log_{\\var{b}}\\left(\\simplify[all,fractionnumbers]{{a_1*a_3^r_2}/{a_2^r_1}}\\right)\\mbox{ on cancelling common factors}.\\end{eqnarray*}\\]</p>\n  <p>Hence $\\displaystyle c=\\simplify[all,fractionnumbers]{{a_1*a_3^r_2}/{a_2^r_1}}$.</p>\n  <p>To calculate $\\displaystyle \\log_{\\var{b}}\\left(\\simplify[all,fractionnumbers]{{a_1*a_3^r_2}/{a_2^r_1}}\\right)$ to 4 decimal places we use the fact that for any positive base $b$:</p>\n  <p>\\[\\log_b(c)=\\frac{\\ln(c)}{\\ln(b)}=\\frac{\\log_{10}(c)}{\\log_{10}(b)}\\]</p>\n  <p>and we can use either of the log functions, $\\ln$ or $\\log_{10}$ on our calculators to find the value.</p>\n  <p>Using $\\ln$ we find:</p>\n  <p>\\[ \\log_{\\var{b}}\\left(\\simplify[all,fractionnumbers]{{a_1*a_3^r_2}/{a_2^r_1}}\\right)=\\frac{\\ln\\left(\\simplify[all,fractionnumbers]{{a_1*a_3^r_2}/{a_2^r_1}}\\right)}{\\ln(\\var{b})}=\\var{ans}\\] to 4 decimal places.</p>\n  ", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"c": {"name": "c", "group": "Ungrouped 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"description": "", "templateType": "anything", "can_override": false}, "tol": {"name": "tol", "group": "Ungrouped variables", "definition": "0.0001", "description": "", "templateType": "anything", "can_override": false}, "ans": {"name": "ans", "group": "Ungrouped variables", "definition": "precround(log(c)/log(b),4)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["c", "b", "a_3", "a_2", "a_1", "r_1", "r_2", "tol", "ans"], "variable_groups": [], "functions": {}, "preamble": {"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>&nbsp;</p>\n<p>Find a fraction or integer $c$ such that:</p>\n<p>$\\log_{\\var{b}}(\\var{a_1})-\\var{r_1}\\log_{\\var{b}}(\\var{a_2})+\\var{r_2}\\log_{\\var{b}}(\\var{a_3})=\\log_{\\var{b}}(c)$&nbsp;</p>\n<p>$c=\\;$[[0]].</p>\n<p>Input $c$ as an integer or as a fraction and not as a decimal.</p>\n<p>Now calculate $\\log_{\\var{b}}(c)$ to 4 decimal places:</p>\n<p>$\\log_{\\var{b}}(c)=\\;$[[1]].</p>\n<p>&nbsp;</p>\n<p>&nbsp;</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{a_1*a_3^r_2}/{a_2^r_1}", "answerSimplification": "all,fractionnumbers", "showPreview": true, 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"maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Wan Mekwi", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4058/"}, {"name": "Ugur Efem", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/14200/"}]}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Wan Mekwi", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4058/"}, {"name": "Ugur Efem", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/14200/"}]}