// Numbas version: exam_results_page_options
{"name": "Using the Logarithm Equivalence  $\\log_ba=c \\Longleftrightarrow a=b^c$", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"rulesets": {}, "variable_groups": [{"variables": ["g3", "g2", "g4", "g1"], "name": "part 2"}, {"variables": ["h1", "h2"], "name": "part3"}], "tags": [], "parts": [{"variableReplacementStrategy": "originalfirst", "customName": "", "showCorrectAnswer": true, "showFeedbackIcon": true, "sortAnswers": false, "variableReplacements": [], "scripts": {}, "marks": 0, "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "useCustomName": false, "gaps": [{"variableReplacementStrategy": "originalfirst", "customName": "", "failureRate": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "vsetRangePoints": 5, "checkingType": "absdiff", "variableReplacements": [], "valuegenerators": [], "scripts": {}, "marks": 1, "extendBaseMarkingAlgorithm": true, "checkingAccuracy": 0.001, "customMarkingAlgorithm": "", "vsetRange": [0, 1], "useCustomName": false, "answer": "{f^f1}", "unitTests": [], "type": "jme", "showPreview": true, "checkVariableNames": false}], "unitTests": [], "type": "gapfill", "prompt": "<p>Rearrange the equation to&nbsp;find $x$.</p>\n<p>$\\log_\\var{f}(x)=\\var{f1}$&nbsp;</p>\n<p>$x=$&nbsp;[[0]]</p>"}, {"variableReplacementStrategy": "originalfirst", "customName": "", "showCorrectAnswer": true, "showFeedbackIcon": true, "sortAnswers": false, "variableReplacements": [], "scripts": {}, "marks": 0, "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "useCustomName": false, "gaps": [{"variableReplacementStrategy": "originalfirst", "customName": "", "failureRate": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "vsetRangePoints": 5, "checkingType": "absdiff", "variableReplacements": [], "valuegenerators": [{"name": "y", "value": ""}], "scripts": {}, "marks": 1, "extendBaseMarkingAlgorithm": true, "checkingAccuracy": 0.001, "customMarkingAlgorithm": "", "vsetRange": [0, 1], "useCustomName": false, "answer": "{g1}^(y+{g2})", "unitTests": [], "type": "jme", "showPreview": true, "checkVariableNames": false}], "unitTests": [], "type": "gapfill", "prompt": "<p>Make $x$ the subject of the following equation.</p>\n<p>$\\log_\\var{g1}(x)=y+\\var{g2}$</p>\n<p>$x=$&nbsp;[[0]]<em><br/></em></p>"}, {"variableReplacementStrategy": "originalfirst", "customName": "", "showCorrectAnswer": true, "showFeedbackIcon": true, "sortAnswers": false, "variableReplacements": [], "scripts": {}, "marks": 0, "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "useCustomName": false, "gaps": [{"variableReplacementStrategy": "originalfirst", "customName": "", "failureRate": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "vsetRangePoints": 5, "checkingType": "absdiff", "variableReplacements": [], "valuegenerators": [{"name": "y", "value": ""}], "scripts": {}, "marks": 1, "extendBaseMarkingAlgorithm": true, "checkingAccuracy": 0.001, "customMarkingAlgorithm": "", "vsetRange": [0, 1], "useCustomName": false, "answer": "(y+{h1})^(1/{h2})", "unitTests": [], "type": "jme", "showPreview": true, "checkVariableNames": false}], "unitTests": [], "type": "gapfill", "prompt": "<p>Make $x$ the subject of the equation, leaving your answer in the form $a^{\\frac{1}{b}}$.</p>\n<p>$\\log_x(y+\\var{h1})=\\var{h2}$</p>\n<p>$x=$&nbsp;[[0]]</p>"}, {"variableReplacementStrategy": "originalfirst", "distractors": ["", "", "", "", "", ""], "maxMarks": 0, "showFeedbackIcon": true, "variableReplacements": [], "useCustomName": false, "marks": 0, "extendBaseMarkingAlgorithm": true, "unitTests": [], "maxAnswers": 0, "matrix": ["1", "1", "1", "-5", "-5", "1"], "type": "m_n_2", "displayColumns": 0, "warningType": "none", "showCorrectAnswer": true, "customName": "", "scripts": {}, "displayType": "checkbox", "customMarkingAlgorithm": "", "showCellAnswerState": true, "choices": ["<p>$\\log_a(a^x)$</p>", "<p>$a^{\\log_a(x)}$</p>", "<p>$e^{\\ln(x)}$</p>", "<p>$\\log_{10}(x)$</p>", "<p>$\\log_e(x)$</p>", "<p>$\\ln(e^x)$</p>"], "minAnswers": 0, "shuffleChoices": true, "prompt": "<p>Which of the following expressions are equivalent to $x$?</p>", "minMarks": 0}], "functions": {}, "name": "Using the Logarithm Equivalence  $\\log_ba=c \\Longleftrightarrow a=b^c$", "ungrouped_variables": ["f", "f2", "f1", "f3", "f4", "f5"], "preamble": {"js": "", "css": ""}, "variablesTest": {"maxRuns": 100, "condition": ""}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Rearrange some expressions involving logarithms by applying the relation&nbsp;$\\log_b(a) = c \\iff a = b^c$.</p>"}, "extensions": [], "statement": "<p>Changing the subject of an equation involving logarithms often requires the use of the equivalence</p>\n<p>\\[\\log_ba=c \\Longleftrightarrow a=b^c\\text{.}\\]</p>", "advice": "<h4>a)</h4>\n<p><em>i)</em></p>\n<p>We can rearrange logarithms using indices.&nbsp;</p>\n<p>\\[\\log_ba=c \\Longleftrightarrow a=b^c\\]</p>\n<p>Using this equivalence we can rewrite&nbsp;$\\log_\\var{f}x=\\var{f1}$.</p>\n<p>\\[\\begin{align}<br/>x&amp;= \\var{f}^\\var{f1} \\\\<br/>&amp;=\\var{f^f1}<br/>\\end{align}\\]</p>\n<p><em></em></p>\n<h4>b)</h4>\n<p><em>i)</em></p>\n<p>We can use the equivalence to rewrite our equation.</p>\n<p>\\[\\log_ba=c \\Longleftrightarrow a=b^c\\]</p>\n<p>We can write out our values to&nbsp;makes it easier.</p>\n<p>\\[\\begin{align}<br/>a&amp;=x \\\\<br/>b&amp;=\\var{g1}\\\\<br/>c&amp;=y+\\var{g2}<br/>\\end{align}\\]</p>\n<p>Then we can write out our equation in the required form.</p>\n<p>\\[x=\\var{g1}^{y+\\var{g2}}\\]</p>\n<p><em></em></p>\n<h4>c)</h4>\n<p>We can&nbsp;use the same equivalence as in part&nbsp;<strong>b)</strong>.&nbsp;</p>\n<p>\\[\\log_ba=c \\Longleftrightarrow a=b^c\\]</p>\n<p>We have</p>\n<p>\\begin{align}<br/>a&amp;=y+\\var{h1} \\\\<br/>b&amp;=x\\\\<br/>c&amp;=\\var{h2}\\text{.} \\\\ \\\\<br/>\\log_{x}(y+\\var{h1}) &amp;= \\var{h2} \\\\<br/>\\implies y+\\var{h1} &amp;= x^{\\var{h2}} \\\\<br/>x &amp;= (y+\\var{h1})^{\\frac{1}{\\var{h2}}}<br/>\\end{align}</p>\n<p></p>\n<h4>d)&nbsp;</h4>\n<p>The two in this list that don't equal $x$ are&nbsp;$\\log_e(x)$ and&nbsp;$\\log_{10}(x)$.</p>\n<p>\\[\\begin{align}<br/>\\log_e(x)&amp;=\\ln(x)\\\\<br/>\\log_{10}(x)&amp;=\\log(x)\\text{.}<br/>\\end{align}\\]</p>", "variables": {"f2": {"description": "", "definition": "random(2..10 except f3 f)", "group": "Ungrouped variables", "name": "f2", "templateType": "anything"}, "f4": {"description": "", "definition": "random(5..12 except f2 f)", "group": "Ungrouped variables", "name": "f4", "templateType": "anything"}, "g4": {"description": "", "definition": "random(2..10except g1 g2 g3)", "group": "part 2", "name": "g4", "templateType": "anything"}, "g1": {"description": "", "definition": "random(2..10)", "group": "part 2", "name": "g1", "templateType": "anything"}, "g3": {"description": "", "definition": "random(2..10except g1 g2)", "group": "part 2", "name": "g3", "templateType": "anything"}, "f3": {"description": "", "definition": "random(3..8)", "group": "Ungrouped variables", "name": "f3", "templateType": "anything"}, "f": {"description": "", "definition": "random(2..10)", "group": "Ungrouped variables", "name": "f", "templateType": "anything"}, "f1": {"description": "", "definition": "random(2..5 except f)", "group": "Ungrouped variables", "name": "f1", "templateType": "anything"}, "h1": {"description": "", "definition": "random(1..10 except h2)", "group": "part3", "name": "h1", "templateType": "anything"}, "g2": {"description": "", "definition": "random(2..10except g1)", "group": "part 2", "name": "g2", "templateType": "anything"}, "f5": {"description": "", "definition": "random(2..6 except f1)", "group": "Ungrouped variables", "name": "f5", "templateType": "anything"}, "h2": {"description": "", "definition": "random(2..4)", "group": "part3", "name": "h2", "templateType": "anything"}}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Hannah Aldous", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1594/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Hannah Aldous", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1594/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}