// Numbas version: finer_feedback_settings
{"name": "Logarithms ", "feedback": {"allowrevealanswer": true, "showtotalmark": true, "advicethreshold": 0, "intro": "", "feedbackmessages": [], "showanswerstate": true, "showactualmark": true, "enterreviewmodeimmediately": true, "showexpectedanswerswhen": "inreview", "showpartfeedbackmessageswhen": "always", "showactualmarkwhen": "always", "showtotalmarkwhen": "always", "showanswerstatewhen": "always", "showadvicewhen": "never"}, "timing": {"allowPause": true, "timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "allQuestions": true, "shuffleQuestions": false, "percentPass": 0, "duration": 0, "pickQuestions": 0, "navigation": {"onleave": {"action": "none", "message": ""}, "reverse": true, "allowregen": true, "showresultspage": "oncompletion", "preventleave": true, "browse": true, "showfrontpage": true}, "metadata": {"description": "<p>Introduction to logs, Rules of logs, Log equations</p>\n<p>rebel</p>\n<p>rebelmaths</p>", "licence": "None specified"}, "type": "exam", "questions": [], "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": [{"name": "What does log mean?", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}], "functions": {}, "ungrouped_variables": ["a", "x", "y"], "tags": ["log", "rebel", "REBEL", "rebelmaths"], "preamble": {"css": "", "js": ""}, "advice": "", "rulesets": {}, "parts": [{"stepsPenalty": 0, "prompt": "<p>$\\log_\\var{a} \\var{x}$ means...</p>\n<p>To what power do I raise [[0]]&nbsp;in order to get [[1]]?</p>\n<p>And the answer is [[2]].</p>\n<p>Hence, &nbsp;$\\log_\\var{a} \\var{x}=$[[3]].</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "{a}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"allowFractions": false, "variableReplacements": [], "maxValue": "x", "minValue": "x", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "{y}", "minValue": "{y}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "{y}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "steps": [{"prompt": "<p style=\"text-align: center;\">Write in the form&nbsp;</p>\n<p style=\"text-align: center;\">$a^?=y$</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "scripts": {}, "marks": 0, "showCorrectAnswer": true, "type": "gapfill"}], "statement": "<p style=\"text-align: center;\">$\\log_a y$ is really asking the question&nbsp;</p>\n<p style=\"text-align: center;\">\"To what power do I raise $a$ in order to get $y$?\"</p>\n<p style=\"text-align: center;\">$a^?=y$</p>\n<p style=\"text-align: left;\">Example:&nbsp;</p>\n<p style=\"text-align: left;\">$\\log_5&nbsp;25$ is really asking the question&nbsp;</p>\n<p style=\"text-align: left;\">\"To what power do I raise $5$ in order to get $25$?\"</p>\n<p style=\"text-align: left;\">$5^?=25$ and we know the answer is 2, that is&nbsp;$\\log_5&nbsp;25=2$ or if you prefer&nbsp;$5^2=25$</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "random(2,3,4,5,10)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "x": {"definition": "a^y ", "templateType": "anything", "group": "Ungrouped variables", "name": "x", "description": ""}, "y": {"definition": "random(2..4 except a)", "templateType": "anything", "group": "Ungrouped variables", "name": "y", "description": ""}}, "metadata": {"notes": "", "description": "", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Converting Logs to Indices", "extensions": [], "custom_part_types": [], "resources": [["question-resources/rules_73.jpg", "/srv/numbas/media/question-resources/rules_73.jpg"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}], "functions": {}, "ungrouped_variables": ["n1", "ans11", "n2", "ans12", "n21", "n31", "ans13", "n32", "n41", "ans21", "n4", "ans22", "n5", "n6", "n7", "n8", "ans23"], "tags": ["logs", "rebel", "rebelmaths"], "advice": "<p>N.B. Log Rules</p>\n<p></p>\n<p>Part 1</p>\n<p>i) $x = \\var{n1[0]}^\\var{n1[1]} = \\var{ans11}$</p>\n<p>ii) $ \\var{n21} = \\var{n2[0]}^x$</p>\n<p>$x = \\var{ans12}$</p>\n<p>iii)&nbsp;$x = \\var{n32}^ \\frac{1}{\\var{n31}} = \\var{ans13}$</p>\n<p>Part 2</p>\n<p>i) $\\frac{1}{\\var{n41}} = \\var{n4[0]}^x&nbsp;$</p>\n<p>$x = \\var{ans21}$</p>\n<p>ii) $\\var{n6} = x^\\var{n5}$</p>\n<p>$x = \\var{ans22}$</p>\n<p>iii)&nbsp;$\\var{n7} = x^ \\frac{1}{\\var{n8}}$</p>\n<p>$ x = \\var{n7}^\\var{n8} = \\var{ans23}$</p>", "rulesets": {}, "parts": [{"stepsPenalty": 0, "prompt": "<p>i) $\\log_\\var{n1[0]} x = \\var{n1[1]}$</p>\n<p>x =&nbsp;[[0]]</p>\n<p>ii) $\\log_\\var{n2[0]} \\var{n21}&nbsp;= x$</p>\n<p>x =&nbsp;[[1]]</p>\n<p>iii) $\\log_\\var{n32} x = \\frac{1}{\\var{n31}}$</p>\n<p>x =&nbsp;[[2]]</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "{ans11}", "minValue": "{ans11}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "{ans12}", "minValue": "{ans12}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "{ans13}", "minValue": "{ans13}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "steps": [{"prompt": "<p>Remember the rule</p>\n<p style=\"text-align: center;\">$a^x=y \\iff \\log_a y=x$</p>\n<p style=\"text-align: left;\">So, for example $5^2=25$ has the same information or is another way of writing the information $\\log_5 25=2$.</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "scripts": {}, "marks": 0, "showCorrectAnswer": true, "type": "gapfill"}, {"stepsPenalty": 0, "prompt": "<p>i) $\\log_\\var{n4[0]} \\frac{1}{\\var{n41}} = x$</p>\n<p>x =&nbsp;[[0]]</p>\n<p>ii) $\\log_x \\var{n6}&nbsp;= \\var{n5}$</p>\n<p>x =&nbsp;[[1]]</p>\n<p>iii) $\\log_x \\var{n7} = \\frac{1}{\\var{n8}}$</p>\n<p>x =&nbsp;[[2]]</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "{ans21}", "minValue": "{ans21}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "{ans22}", "minValue": "{ans22}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "{ans23}", "minValue": "{ans23}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "steps": [{"prompt": "<p>Remember the rule</p>\n<p style=\"text-align: center;\">$a^x=y \\iff \\log_a y=x$</p>\n<p style=\"text-align: left;\">So, for example $5^2=25$ has the same information or is another way of writing the information $\\log_5 25=2$.</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "scripts": {}, "marks": 0, "showCorrectAnswer": true, "type": "gapfill"}], "statement": "", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"n21": {"definition": "n2[0]^ans12", "templateType": "anything", "group": "Ungrouped variables", "name": "n21", "description": ""}, "ans12": {"definition": "random(2..3)", "templateType": "anything", "group": "Ungrouped variables", "name": "ans12", "description": ""}, "n2": {"definition": "shuffle(3..7)[0..2]", "templateType": "anything", "group": "Ungrouped variables", "name": "n2", "description": ""}, "n41": {"definition": "n4[0]^-ans21", "templateType": "anything", "group": "Ungrouped variables", "name": "n41", "description": ""}, "n4": {"definition": "shuffle(2..5)[0..2]", "templateType": "anything", "group": "Ungrouped variables", "name": "n4", "description": ""}, "ans22": {"definition": "random(3..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "ans22", "description": ""}, "n8": {"definition": "random(2..4)", "templateType": "anything", "group": "Ungrouped variables", "name": "n8", "description": ""}, "ans13": {"definition": "n2[1]", "templateType": "anything", "group": "Ungrouped variables", "name": "ans13", "description": ""}, "ans11": {"definition": "n1[0]^n1[1]", "templateType": "anything", "group": "Ungrouped variables", "name": "ans11", "description": ""}, "ans21": {"definition": "n4[1]*-1", "templateType": "anything", "group": "Ungrouped variables", "name": "ans21", "description": ""}, "n1": {"definition": "shuffle(2..5)[0..2]", "templateType": "anything", "group": "Ungrouped variables", "name": "n1", "description": ""}, "n31": {"definition": "random(2..3)", "templateType": "anything", "group": "Ungrouped variables", "name": "n31", "description": ""}, "n32": {"definition": "ans13^n31", "templateType": "anything", "group": "Ungrouped variables", "name": "n32", "description": ""}, "ans23": {"definition": "n7^n8", "templateType": "anything", "group": "Ungrouped variables", "name": "ans23", "description": ""}, "n5": {"definition": "random(2..4)", "templateType": "anything", "group": "Ungrouped variables", "name": "n5", "description": ""}, "n6": {"definition": "ans22^n5", "templateType": "anything", "group": "Ungrouped variables", "name": "n6", "description": ""}, "n7": {"definition": "random(2..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "n7", "description": ""}}, "metadata": {"notes": "<p>rebelmaths rebel Rebel REBEL</p>", "description": "<p>Practice converting from logarithm notation to index notation and hence solving simple equations.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Adding and Subtracting Logs", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}], "functions": {}, "ungrouped_variables": ["a1", "a2", "b1", "b2", "c1", "c2", "c3", "d1", "d2", "d3", "e1", "e2", "e3", "f1", "f2", "f3", "f4", "f5", "g1", "g2", "g3", "g4", "h1", "h2", "h3", "i1", "i2", "i3", "i4"], "tags": ["Logarithm", "Logs", "rebel", "rebelmaths", "Rules of Logs"], "preamble": {"css": "", "js": ""}, "advice": "<p>Use of the laws of logarithms is crucial here:</p>\n<p>$\\log{a} + \\log{b} = \\log{ab}$</p>\n<p>$\\log{a} - \\log{b} = \\log{\\frac{a}{b}}$</p>\n<p>$\\log{a^n} = n\\log{a}$</p>\n<p></p>", "rulesets": {}, "parts": [{"prompt": "<p>$\\log{\\var{a1}} + \\log{\\var{a2}} = \\log$[[0]]</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "a1*a2", "minValue": "a1*a2", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<p>$\\log{\\var{b1}} + \\log{\\var{b2}} = \\log$[[0]]</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "b1*b2", "minValue": "b1*b2", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<p>$\\log{\\var{c3}} - \\log{\\var{c2}} = \\log$[[0]]</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "c1", "minValue": "c1", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<p>$\\log{\\var{e1}} + \\log{\\var{e2}} + \\log{\\var{e3}} = \\log$[[0]]</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "e1*e2*e3", "minValue": "e1*e2*e3", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<p>$\\log{\\var{f1}} + \\log{\\var{f3}} - \\log{\\var{f5}} = \\log$[[0]]</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "f1*f3/f5", "minValue": "f1*f3/f5", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<p>$\\var{g1}\\log{\\var{g2}} + \\log{\\var{g3}} - \\log{\\var{g4}}=\\log$[[0]]</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "g2^(g1-1)", "minValue": "g2^(g1-1)", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<p>$\\frac{1}{2}\\log{\\var{h3}} - \\frac{1}{2}\\log{\\var{h2}}=\\log$[[0]]</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "(h1)^0.5", "minValue": "(h1)^0.5", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<p>$\\log{\\var{i3}}-\\frac{1}{2}\\log{\\var{i4}} = \\log$[[0]]</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "i1", "minValue": "i1", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "statement": "<p>Express each of the following as a single logarithm</p>\n<p></p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"f1": {"definition": "random(2..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "f1", "description": ""}, "f2": {"definition": "random(3..5 except f1)", "templateType": "anything", "group": "Ungrouped variables", "name": "f2", "description": ""}, "f3": {"definition": "f1*f2", "templateType": "anything", "group": "Ungrouped variables", "name": "f3", "description": ""}, "f4": {"definition": "f2-1", "templateType": "anything", "group": "Ungrouped variables", "name": "f4", "description": ""}, "f5": {"definition": "f1^2", "templateType": "anything", "group": "Ungrouped variables", "name": "f5", "description": ""}, "h2": {"definition": "random([2,3,5,7,8,10])", "templateType": "anything", "group": "Ungrouped variables", "name": "h2", "description": ""}, "h3": {"definition": "h1*h2", "templateType": "anything", "group": "Ungrouped variables", "name": "h3", "description": ""}, "h1": {"definition": "random([4,9,16,25])", "templateType": "anything", "group": "Ungrouped variables", "name": "h1", "description": ""}, "b1": {"definition": "random(2..19 except a2 except a1)", "templateType": "anything", "group": "Ungrouped variables", "name": "b1", "description": ""}, "b2": {"definition": "random(2..19 except a1 except a2 except b1)", "templateType": "anything", "group": "Ungrouped variables", "name": "b2", "description": ""}, "d2": {"definition": "random(2..9 except d1)", "templateType": "anything", "group": "Ungrouped variables", "name": "d2", "description": ""}, "d3": {"definition": "d1*d2", "templateType": "anything", "group": "Ungrouped variables", "name": "d3", "description": ""}, "d1": {"definition": "random(2..19)", "templateType": "anything", "group": "Ungrouped variables", "name": "d1", "description": ""}, "g4": {"definition": "g2*g3", "templateType": "anything", "group": "Ungrouped variables", "name": "g4", "description": ""}, "g3": {"definition": "random(2..9 except g2 except g1)", "templateType": "anything", "group": "Ungrouped variables", "name": "g3", "description": ""}, "g2": {"definition": "random(2..9 except g1)", "templateType": "anything", "group": "Ungrouped variables", "name": "g2", "description": ""}, "g1": {"definition": "random(2..4)", "templateType": "anything", "group": "Ungrouped variables", "name": "g1", "description": ""}, "i1": {"definition": "random(2..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "i1", "description": ""}, "i3": {"definition": "i1*i2", "templateType": "anything", "group": "Ungrouped variables", "name": "i3", "description": ""}, "i2": {"definition": "random(2..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "i2", "description": ""}, "i4": {"definition": "i2^2", "templateType": "anything", "group": "Ungrouped variables", "name": "i4", "description": ""}, "a1": {"definition": "random(2..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "a1", "description": ""}, "a2": {"definition": "random(2..9 except a1)", "templateType": "anything", "group": "Ungrouped variables", "name": "a2", "description": ""}, "c3": {"definition": "c1*c2", "templateType": "anything", "group": "Ungrouped variables", "name": "c3", "description": ""}, "c2": {"definition": "random(2..9 except c1)", "templateType": "anything", "group": "Ungrouped variables", "name": "c2", "description": ""}, "c1": {"definition": "random(2..19)", "templateType": "anything", "group": "Ungrouped variables", "name": "c1", "description": ""}, "e1": {"definition": "random(2..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "e1", "description": ""}, "e3": {"definition": "random(2..9 except e1 except e2)", "templateType": "anything", "group": "Ungrouped variables", "name": "e3", "description": ""}, "e2": {"definition": "random(2..9 except e1)", "templateType": "anything", "group": "Ungrouped variables", "name": "e2", "description": ""}}, "metadata": {"notes": "<p>rebelmaths rebel Rebel REBEL</p>", "description": "<p>Practice using the log rules to add and subtract logarithms</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Logarithms: Solving equations 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}], "functions": {}, "ungrouped_variables": ["a", "c", "b", "d"], "tags": ["algebra", "algebraic manipulation", "combining logarithms", "logarithm laws", "logarithms", "rebel", "rebelmaths", "simplifying logarithms", "solving equations", "Solving equations", "steps", "Steps"], "preamble": {"css": "", "js": ""}, "advice": "<p>We use the following two rules for logs :</p>\n<p>1.&nbsp;$\\log_a(\\frac{x}{y})=\\log_a(x)-\\log_a(y)$</p>\n<p>2.&nbsp;$a^x=y \\iff \\log_a y=x$</p>\n<p>Using rule 1 we get<br/>\\[\\log_{\\var{a}}(x+\\var{b})- \\log_{\\var{a}}(\\simplify{(x+{c})})=\\log_{\\var{a}}\\left(\\simplify{(x+{b})/(x+{c})}\\right)\\]<br/>So the equation to solve becomes:<br/>\\[\\log_{\\var{a}}\\left(\\simplify{(x+{b})/(x+{c})}\\right)=\\var{d}\\]<br/>and using rule 2 this gives:<br/>\\[ \\begin{eqnarray} \\simplify{(x+{b})/(x+{c})}&amp;=&amp;{\\var{a}}^{\\var{d}}\\Rightarrow\\\\ x+\\var{b}&amp;=&amp;{\\var{a}}^{\\var{d}}(x+\\var{c})=\\simplify{{a^d}}(x+\\var{c})\\Rightarrow\\\\ \\simplify{{a^d-1}x}&amp;=&amp;\\simplify[std]{{b}-{c}*{a^d}={b-c*a^d}}\\Rightarrow\\\\ x&amp;=&amp;\\simplify{{b-c*a^d}/{a^d-1}} \\end{eqnarray} \\]<br/>We should check that this solution gives positive values for $x+\\var{b}$ and $\\simplify{x+{c}}$ as otherwise the logs are not defined.</p>\n<p>Substituting this value for $x$ into $\\log_{\\var{a}}(x+\\var{b})$ we get $\\log_{\\var{a}}(\\simplify{({b-c }{a^d})/{a^d-1}})$ so OK.</p>\n<p>For $\\log_{\\var{a}}(\\simplify{x+{c}})$ we get on substituting for $x$, $\\log_{\\var{a}}(\\simplify{({b-c })/{a^d-1}})$ so OK.</p>\n<p>Hence the value we found for $x$ is a solution to the original equation.</p>", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "parts": [{"stepsPenalty": 1, "prompt": "\n      <p>\\[\\log_{\\var{a}}(x+\\var{b})- \\log_{\\var{a}}(\\simplify{(x+{c})})=\\var{d}\\]</p>\n      <p>$x=\\;$ [[0]]</p>\n      <p>If you want help in solving the equation, click on Show steps. If you do so then you will lose 1 mark.</p>\n      <p>Input all numbers as fractions or integers and not as decimals.</p>\n      ", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"notallowed": {"message": "<p>Input as a fraction or an integer, not as a decimal.</p>", "showStrings": false, "strings": ["."], "partialCredit": 0}, "vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.0001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "{b-c*a^d}/{a^d-1}", "marks": 2, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "steps": [{"prompt": "<p>Two rules for logs should be used:</p>\n<p>1.&nbsp;$\\log_a(\\frac{x}{y})=\\log_a(x)-\\log_a(y)$</p>\n<p>2.&nbsp;$a^x=y \\iff \\log_a y=x$</p>\n<p>Use rule 1 followed by rule 2 to get an equation for $x$.</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "scripts": {}, "marks": 0, "showCorrectAnswer": true, "type": "gapfill"}], "statement": "\n  <p>Solve the following equation for $x$.</p>\n  <p>Input your answer as a fraction or an integer as appropriate and not&nbsp;as a decimal.</p>\n  ", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "random(2..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "b-random(1..20)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "random(1..20)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "d": {"definition": "random(1,2)", "templateType": "anything", "group": "Ungrouped variables", "name": "d", "description": ""}}, "metadata": {"notes": "<p><strong>5/08/2012:</strong></p>\n<p>Added tags.</p>\n<p>Added description.</p>\n<p>Checked calculation.OK.</p>\n<p>Improved display in content areas.</p>\n<p>rebelmaths rebel Rebel REBEL</p>", "description": "<p>Solve for $x$: &nbsp;$\\log_{a}(x+b)- \\log_{a}(x+c)=d$</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Logarithms: Solving equations 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}], "functions": {}, "ungrouped_variables": ["a", "c", "b", "d", "s", "sol2", "sol1"], "tags": ["algebra", "algebraic manipulation", "combining logarithms", "logarithm laws", "logarithms", "rebel", "rebelmaths", "simplifying logarithms", "solving", "solving equations", "Solving equations", "steps", "Steps"], "preamble": {"css": "", "js": ""}, "advice": "<p>We use the following rules for logs:</p>\n<p>1.&nbsp;$\\log_a(x^q)=q\\log_a(x)$</p>\n<p>2. $\\log_a(\\frac{x}{y})=\\log_a(x)-\\log_a(y)$</p>\n<p>3.&nbsp;$a^x=y \\iff \\log_a y=x$</p>\n<p>Using rule 1 we get<br/>\\[2\\log_{\\var{a}}(\\simplify{x+{b}})- \\log_{\\var{a}}(\\simplify{(x+{c})})=\\log_{\\var{a}}((\\simplify{x+{b}})^2)- \\log_{\\var{a}}(\\simplify{(x+{c})})\\]<br/>Using rule 2 gives<br/>\\[\\log_{\\var{a}}(\\simplify{(x+{b})^2})- \\log_{\\var{a}}(\\simplify{(x+{c})})=\\log_{\\var{a}}\\left(\\simplify{(x+{b})^2/(x+{c})}\\right)\\]<br/>So the equation to solve becomes:<br/>\\[\\log_{\\var{a}}\\left(\\simplify{(x+{b})^2/(x+{c})}\\right)=\\var{d}\\]<br/>and using rule 3 this gives:<br/>\\[ \\begin{eqnarray} \\simplify{(x+{b})^2/(x+{c})}&amp;=&amp;{\\var{a}}^{\\var{d}}\\Rightarrow\\\\ \\simplify{(x+{b})^2}&amp;=&amp;{\\var{a}}^{\\var{d}}(\\simplify{x+{c}})=\\simplify{{a^d}(x+{c})}\\Rightarrow\\\\ \\simplify{x^2+{2*b-a^(d)}x+{b^2-a^(d)*c}}&amp;=&amp;0 \\end{eqnarray} \\]<br/>Solving this quadratic we get two solutions:</p>\n<p>$x=\\var{sol1}$ and $x=\\var{sol2}$</p>\n<p>We should check that these solutions gives positive values for $\\simplify{x+{b}}$ and $\\simplify{x+{c}}$ as otherwise the logs are not defined.</p>\n<p><strong>The value $x=\\var{sol1}$ gives:&nbsp;</strong></p>\n<p>Substituting this value for $x$ into $\\log_{\\var{a}}(\\simplify{x+{b}})$ we get $\\log_{\\var{a}}(\\simplify{{2*a^d}})$ so OK.</p>\n<p>Substituting this value for $x$ into $\\log_{\\var{a}}(\\simplify{x+{c}})$ we get $\\log_{\\var{a}}(\\simplify{{4*a^d}})$ so OK.</p>\n<p>Hence $x=\\var{sol1}$ is a solution to our original equation.</p>\n<p><strong>The value $x=\\var{sol2}$ gives:</strong></p>\n<p>Substituting this value for $x$ into $\\log_{\\var{a}}(\\simplify{x+{b}})$ we get $\\log_{\\var{a}}(\\simplify{{-a^d}})$ so <span class=\"caps\">NOT</span> OK.</p>\n<p>Substituting this value for $x$ into $\\log_{\\var{a}}(\\simplify{x+{c}})$ we get $\\log_{\\var{a}}(\\simplify{{a^d}})$ so OK.</p>\n<p>Hence $x=\\var{sol2}$ is <span class=\"caps\">NOT</span> a solution to our original equation as $\\log_{\\var{a}}(\\simplify{x+{b}})$ is not defined for this value of $x$.</p>\n<p>So there is only one solution $x=\\var{sol1}$.</p>", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"]}, "parts": [{"stepsPenalty": 1, "prompt": "\n      <p>\\[2\\log_{\\var{a}}(\\simplify{x+{b}})- \\log_{\\var{a}}(\\simplify{(x+{c})})=\\var{d}\\]</p>\n      <p>$x=\\;$ [[0]].</p>\n      <p>If you want help in solving the equation, click on Show steps. If you do so then you will lose 1 mark.</p>\n      <p>Input all numbers as fractions or integers and not as decimals.</p>\n      ", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"notallowed": {"message": "<p>Input as an integer, not as a decimal.</p>", "showStrings": false, "strings": ["."], "partialCredit": 0}, "vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.0001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "{sol1}", "marks": 2, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "steps": [{"prompt": "<p>Three rules for logs should be used:</p>\n<p>1.&nbsp;$\\log_a(x^q)=q\\log_a(x)$</p>\n<p>2. $\\log_a(\\frac{x}{y})=\\log_a(x)-\\log_a(y)$</p>\n<p>3.&nbsp;$a^x=y \\iff \\log_a y=x$</p>\n<p>So use rule 1 followed by rules 2 and 3 to get an equation for $x$.</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "scripts": {}, "marks": 0, "showCorrectAnswer": true, "type": "gapfill"}], "statement": "\n  <p>Solve the following equation for $x$.</p>\n  <p>Input your answer as a fraction or an integer as appropriate and not&nbsp;as a decimal.</p>\n  ", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "random(2,3)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "b+2*a^(d)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "s*random(1..20)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "d": {"definition": "random(1,2)", "templateType": "anything", "group": "Ungrouped variables", "name": "d", "description": ""}, "s": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s", "description": ""}, "sol2": {"definition": "-c+a^d", "templateType": "anything", "group": "Ungrouped variables", "name": "sol2", "description": ""}, "sol1": {"definition": "c-2*b", "templateType": "anything", "group": "Ungrouped variables", "name": "sol1", "description": ""}}, "metadata": {"notes": "<p><strong>5/08/2012:</strong></p>\n<p>Added tags.</p>\n<p>Added description.</p>\n<p>Checked calculation.OK.</p>\n<p>Improved display in content areas.</p>\n<p>rebelmaths rebel Rebel REBEL</p>", "description": "\n            \t\t<p>Solve for $x$: $\\displaystyle 2\\log_{a}(x+b)- \\log_{a}(x+c)=d$.&nbsp;</p>\n            \t\t        <p>Make sure that your choice is a solution by substituting back into the equation.</p>\n            \t\t", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Logarithms: Solving equations 3", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "tags": ["exponential", "exponentiation", "laws of logarithms", "laws of logs", "log laws", "logarithm laws", "logarithm rules", "logarithms", "logs", "solving equations", "solving logarithmic equations"], "advice": "\n    <p>First use one of the logarith laws which states (for logarithms to any base)</p>\n    <p>\\[\\log(a)-\\log(b)=\\log\\left(\\frac{a}{b}\\right)\\]</p>\n    <p>So the equation can be written as:</p>\n    <p>\\[\\log_{10}\\left(\\simplify[std]{({a1}x+{b1})/({c1}x+{d1})}\\right)=\\var{e1}\\]<br/>Now exponentiate both sides to get:<br/>\\[\\simplify[std,!otherNumbers]{({a1}x+{b1})/({c1}x+{d1})}=10^{\\var{e1}} \\Rightarrow \\simplify[std,!otherNumbers]{{a1}x+{b1}=10^{e1}({c1}x+{d1})}\\]<br/>Collect together terms in $x$ on the left and everything else on the right of the equation gives:<br/>\\[\\simplify[std,!otherNumbers]{x({a1}-10^{e1}*{c1})=10^{e1}*{d1}-{b1}}\\]<br/>Finally rearrange to get:<br/>\\[\\simplify[std]{x=(10^{e1}*{d1}-{b1})/({a1}-10^{e1}*{c1})={10^e1*d1-b1}/{a1-10^e1*c1}}\\]<br/>which to 3 decimal places evaluates to <br/>\\[x=\\var{ans}.\\]</p>\n    ", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"prompt": "\n        <p>Input the solution for $x$ here:</p>\n      <p>$x=\\;\\;$ [[0]]</p>\n      <p>Input your answer to 3 decimal places.</p>\n        ", "gaps": [{"minvalue": "ans-tol", "type": "numberentry", "maxvalue": "ans+tol", "marks": 1.0, "showPrecisionHint": false}], "type": "gapfill", "marks": 0.0}], "statement": "\n    <p>Solve the following equation for $x$.</p>\n  <p>\\[\\simplify[std]{log({a1}x+{b1})-log({c1}x+{d1})={e1}}\\]</p>\n    ", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"e1": {"definition": "random(1..2)", "name": "e1"}, "a1": {"definition": "random(1..9)", "name": "a1"}, "b1": {"definition": "9+random(1..9)", "name": "b1"}, "tol": {"definition": 0.0, "name": "tol"}, "ans": {"definition": "precround(tans,3)", "name": "ans"}, "c1": {"definition": "d1*random(1..9)", "name": "c1"}, "tans": {"definition": "(d1*10^e1-b1)/(a1-c1*10^e1)", "name": "tans"}, "d1": {"definition": "random(1..9)", "name": "d1"}}, "metadata": {"notes": "\n                \t\t<p><strong>2/07/2012:</strong></p>\n                \t\t<p>Added tags.</p>\n                \t\t<p>Solution to 3 decimal places - no tolerance via new tolerance variable tol=0.</p>\n                \t\t<p>Improved display of Advice.</p>\n                \t\t<p><strong>19/07/2012:</strong></p>\n                \t\t<p>Added description.</p>\n                \t\t<p>Checked calculation.</p>\n                \t\t<p>&nbsp;</p>\n                \t\t<p><strong>25/07/2012:</strong></p>\n                \t\t<p>Added tags.</p>\n                \t\t<p>Removed a stray full stop.</p>\n                \t\t<p>Question appears to be working correctly.</p>\n                \t\t<p>&nbsp;</p>\n                \t\t", "description": "<p>Solve for $x$: $\\log(ax+b)-\\log(cx+d)=s$</p>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Logarithms: Solving equations 4", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "tags": ["equations solved by using logarithms", "laws of logarithms", "logarithm laws", "logarithm rules", "logarithmic expressions", "logarithms", "solving equations by taking logarithms", "solving logarithmic equations"], "advice": "<p>Both parts of this question can be solved in a similar way, by taking logarithms of both sides of each equation.</p>\n  <p>a)</p>\n  <p>Taking logs (to the base 10 in this case &ndash; but any base will do) of both sides of</p>\n  <p>\\[\\var{n}^{\\simplify{{a1}*x+{b1}}}=\\var{m}^{\\var{c1}x}\\]</p>\n  <p>gives on using the rule $\\log(a^b)=b\\log(a)$:</p>\n  <p>\\[\\begin{eqnarray*} \\simplify[std]{({a1}x+{b1})log({n})}&amp;=&amp;\\simplify[std]{{c1}*x*log({m})}\\\\ \\Rightarrow\\simplify[std]{x({a1}*log({n})-{c1}*log({m}))} &amp;=&amp;\\var{-b1}\\log(\\var{n})\\\\ \\Rightarrow x&amp;=&amp;\\simplify[std]{({-b1}*log({n}))/({a1}log({n})-{c1}*log({m}))}\\\\ &amp;=&amp; \\var{ans1}\\mbox{ to 3 decimal places} \\end{eqnarray*} \\]</p>\n  <p>b)</p>\n  <p>Similarly, taking logs of both sides of:<br/>\\[\\var{a2}^{\\var{b2}x^2}=\\var{c2}^{\\var{d2}x}\\]</p>\n  <p>gives:</p>\n  <p>\\[ \\simplify[std]{({b2}x^2)log({a2})}=\\simplify[std]{{d2}*x*log({c2})} \\Rightarrow \\simplify[std]{x({b2}*log({a2})x-{d2}*log({c2}))} =0\\]</p>\n  <p>and so the solutions are:</p>\n  <p>1. $x=0$</p>\n  <p>or</p>\n  <p>2. $\\displaystyle x=\\simplify[std,!fractionNumbers]{({d2}*log({c2}))/({b2}*log({a2})) = {ans2}}$ to 3 decimal places.</p>", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"prompt": "\n        <p>\\[\\var{n}^{\\simplify{{a1}*x+{b1}}}=\\var{m}^{\\var{c1}x}\\]</p>\n        <p>$x=\\;\\;$[[0]].</p>\n        <p>Enter your answer to 3 decimal places.</p>\n        ", "gaps": [{"minvalue": "{ans1}", "type": "numberentry", "maxvalue": "{ans1}", "marks": 1.0, "showPrecisionHint": false}], "type": "gapfill", "marks": 0.0}, {"prompt": "\n        <p>\\[\\var{a2}^{\\var{b2}x^2}=\\var{c2}^{\\var{d2}x}\\]</p>\n        <p>$x=\\;\\;$[[0]] $\\;\\;\\;$ (Enter the smallest value of $x$ here).</p>\n        <p>$x=\\;\\;$[[1]] $\\;\\;\\;$ (Enter the largest value of $x$ here).</p>\n        <p>Enter your answers to 3 decimal places.</p>\n        ", "gaps": [{"minvalue": 0.0, "type": "numberentry", "maxvalue": 0.0, "marks": 0.5, "showPrecisionHint": false}, {"minvalue": "ans2", "type": "numberentry", "maxvalue": "ans2", "marks": 1.5, "showPrecisionHint": false}], "type": "gapfill", "marks": 0.0}], "statement": "<p>Find all values of $x$ that satisfy the following equations:</p>", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"tc2": {"definition": "random(2..9)", "name": "tc2"}, "s3": {"definition": "random(1,-1)", "name": "s3"}, "ans1": {"definition": "precround(tans1,3)", "name": "ans1"}, "s1": {"definition": "random(1,-1)", "name": "s1"}, "m": {"definition": "if(tm=n,17,tm)", "name": "m"}, "d2": {"definition": "random(2..9)", "name": "d2"}, "n": {"definition": "random(2,3,5,7,11,13)", "name": "n"}, "a1": {"definition": "s1*random(1..9)", "name": "a1"}, "tm": {"definition": "random(2,3,5,7,11,13)", "name": "tm"}, "b1": {"definition": "random(1..9)", "name": "b1"}, "b2": {"definition": "random(2..9)", "name": "b2"}, "c2": {"definition": "if(tc2=a2,tc2+1,tc2)", "name": "c2"}, "c1": {"definition": "s3*random(2..9)", "name": "c1"}, "tans1": {"definition": "-b1*log(n)/(a1*log(n)-c1*log(m))", "name": "tans1"}, "ans2": {"definition": "precround(tans2,3)", "name": "ans2"}, "a2": {"definition": "random(2..9)", "name": "a2"}, "tans2": {"definition": "d2*log(c2)/(b2*log(a2))", "name": "tans2"}}, "metadata": {"notes": "\n                        \t\t<p><strong>2/06/2012:</strong></p>\n                        \t\t<p>Added tags.</p>\n                        \t\t<p>Improved display.</p>\n                        \t\t<p>Forced solution in second part to be accurate to 3 decimal places with no tolerance.</p>\n                        \t\t<p>&nbsp;<strong>19/07/2012:</strong></p>\n                        \t\t<p>Added description.</p>\n                        \t\t<p>Checked calculation.</p>\n                        \t\t<p><strong>25/07/2012:</strong></p>\n                        \t\t<p>Added tags.</p>\n                        \t\t<p>Corrected a typo.</p>\n                        \t\t<p>In the Advice section moved \\Rightarrow so that it is at the beginning of the line instead of the end of the previous line.</p>\n                        \t\t<p>Question appears to be working correctly.</p>\n                        \t\t<p>&nbsp;</p>\n                        \t\t<p>&nbsp;</p>\n                        \t\t", "description": "<p>Solve for $x$ each of the following equations:&nbsp;$n^{ax+b}=m^{cx}$ and $p^{rx^2}=q^{sx}$.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Simple equations involving indices", "extensions": [], "custom_part_types": [], "resources": [["question-resources/rules_73.jpg", "/srv/numbas/media/question-resources/rules_73.jpg"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}], "functions": {}, "ungrouped_variables": ["n1", "ans11", "n2", "ans12", "n21", "n31", "ans13", "n32", "n41", "ans21", "n4", "ans22", "n5", "n6", "n7", "n8", "ans23", "an1", "an11", "n10", "an12", "n11", "n12", "b", "c", "na1", "v1", "v2", "v3", "v4", "v5", "v60", "v61", "v7", "ans24"], "tags": ["rebel", "rebelmaths"], "advice": "<p></p>\n<p>i) $\\var{n4[0]} + 10^{\\var{n6}-x}&nbsp;= \\var{v1}$</p>\n<p>$ 10^{\\var{n6}-x}&nbsp;= \\var{v1}-\\var{n4[0]} $</p>\n<p>Taking $\\log$ of both sides,</p>\n<p>$\\var{n6}-x = &nbsp;\\log(\\var{v1}-\\var{n4[0]})$</p>\n<p>$\\var{n6} - \\log(\\var{v1}-\\var{n4[0]})=x$</p>\n<p>$x = \\var{n6} - \\log(\\var{v1}-\\var{n4[0]}) = \\var{ans21}$</p>\n<p></p>\n<p>ii)&nbsp;$\\var{n4[1]}(10^{\\var{v2}x-\\var{v3}} )= \\var{v4}$</p>\n<p>$10^{\\var{v2}x-\\var{v3}} = \\frac{\\var{v4}}{\\var{n4[1]}}$</p>\n<p>Take $\\log$ of both sides,</p>\n<p>$\\var{v2}x -\\var{v3}= &nbsp;\\log \\left( \\frac{\\var{v4}}{\\var{n4[1]}}\\right)$</p>\n<p>$\\var{v2}x = \\var{v3} +\\log \\left( \\frac{\\var{v4}}{\\var{n4[1]}}\\right)$</p>\n<p>$x = \\frac{\\var{v3} +\\log \\left( \\frac{\\var{v4}}{\\var{n4[1]}}\\right)}{ &nbsp;\\var{v2}} = \\var{ans22}$</p>\n<p></p>\n<p></p>\n<p>iii)&nbsp;$\\var{v5[1]}e^{-\\var{v5[0]}x} = \\var{v5[2]}$</p>\n<p>$e^{-\\var{v5[0]}x} = \\frac{\\var{v5[2]}}{\\var{v5[1]}}$</p>\n<p>Take $\\ln$ of both sides,</p>\n<p>$-\\var{v5[0]}x = \\ln(\\frac{\\var{v5[2]}}{\\var{v5[1]}})$</p>\n<p>$x = \\frac{\\ln(\\frac{\\var{v5[2]}}{\\var{v5[1]}})}{ -\\var{v5[0]}} = \\var{ans23}$</p>\n<p></p>\n<p></p>\n<p></p>\n<p>iv)&nbsp;$\\var{v60} = \\var{v61}(1-e^{\\frac{-\\var{v7[0]}x}{\\var{v7[1]}}})$</p>\n<p>$\\frac{\\var{v60} }{ \\var{v61}}=1-e^{\\frac{-\\var{v7[0]}x}{\\var{v7[1]}}}$</p>\n<p>$e^{\\frac{-\\var{v7[0]}x}{\\var{v7[1]}}}=1-\\frac{\\var{v60} }{ \\var{v61}}$</p>\n<p></p>\n<p>Take $\\ln$ of both sides,</p>\n<p>$\\frac{-\\var{v7[0]}x}{\\var{v7[1]}}=\\ln(1-\\frac{\\var{v60} }{ \\var{v61}})$</p>\n<p>$x = (\\frac{\\var{v7[1]}}{-\\var{v7[0]}}) &nbsp;\\ln(1-(\\frac{\\var{v60}}{\\var{v61}}))&nbsp;= \\var{ans24}$</p>", "rulesets": {}, "parts": [{"prompt": "<p>i) $\\var{n4[0]} + 10^{\\var{n6}-x}&nbsp;= \\var{v1}$</p>\n<p>x =&nbsp;[[0]]</p>\n<p>ii)&nbsp;$\\var{n4[1]}(10^{\\var{v2}x-\\var{v3}} )= \\var{v4}$</p>\n<p>x =&nbsp;[[1]]</p>\n<p>iii)&nbsp;$\\var{v5[1]}e^{-\\var{v5[0]}x} = \\var{v5[2]}$</p>\n<p>x =&nbsp;[[2]]</p>\n<p>iv)&nbsp;$\\var{v60} = \\var{v61}(1-e^{\\frac{-\\var{v7[0]}x}{\\var{v7[1]}}})$</p>\n<p>x =&nbsp;[[3]]</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "precision": "2", "maxValue": "{ans21}", "variableReplacementStrategy": "originalfirst", "strictPrecision": false, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": 0, "scripts": {}, "marks": 1, "minValue": "{ans21}", "type": "numberentry", "showPrecisionHint": false}, {"precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "precision": "2", "maxValue": "{ans22}", "variableReplacementStrategy": "originalfirst", "strictPrecision": false, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": 0, "scripts": {}, "marks": 1, "minValue": "{ans22}", "type": "numberentry", "showPrecisionHint": false}, {"precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "precision": "2", "maxValue": "{ans23}", "variableReplacementStrategy": "originalfirst", "strictPrecision": false, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": 0, "scripts": {}, "marks": 1, "minValue": "{ans23}", "type": "numberentry", "showPrecisionHint": false}, {"precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "precision": "2", "maxValue": "{ans24}", "variableReplacementStrategy": "originalfirst", "strictPrecision": false, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": 0, "scripts": {}, "marks": 1, "minValue": "{ans24}", "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "statement": "<p>Solve for $x$ in the following, correct to 2 decimal places:&nbsp;</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"n12": {"definition": "shuffle(2..8)[1..6]", "templateType": "anything", "group": "Ungrouped variables", "name": "n12", "description": ""}, "n10": {"definition": "shuffle(2..8)[1..4]", "templateType": "anything", "group": "Ungrouped variables", "name": "n10", "description": ""}, "n11": {"definition": "random(1..3)", "templateType": "anything", "group": "Ungrouped variables", "name": "n11", "description": ""}, "n31": {"definition": "random(2..3)", "templateType": "anything", "group": "Ungrouped variables", "name": "n31", "description": ""}, "n32": {"definition": "ans13^n31", "templateType": "anything", "group": "Ungrouped variables", "name": "n32", "description": ""}, "ans24": {"definition": "(v7[1]/-v7[0])*ln(1-(v60/v61))", "templateType": "anything", "group": "Ungrouped variables", "name": "ans24", "description": ""}, "ans23": {"definition": "(ln(v5[2]/v5[1]))/-v5[0]", "templateType": "anything", "group": "Ungrouped variables", "name": "ans23", "description": ""}, "ans22": {"definition": "(v3 +log(v4/n4[1]))/v2", "templateType": "anything", "group": "Ungrouped variables", "name": "ans22", "description": ""}, "ans21": {"definition": "n6 - log(v1-n4[0])", "templateType": "anything", "group": "Ungrouped variables", "name": "ans21", "description": ""}, "na1": {"definition": "shuffle(2..4)[0..2]", "templateType": "anything", "group": "Ungrouped variables", "name": "na1", "description": ""}, "an12": {"definition": "n10[1]-(an1*n11)", "templateType": "anything", "group": "Ungrouped variables", "name": "an12", "description": ""}, "an11": {"definition": "n10[0]-(an1*n10[2])", "templateType": "anything", "group": "Ungrouped variables", "name": "an11", "description": ""}, "n41": {"definition": "1", "templateType": "anything", "group": "Ungrouped variables", "name": "n41", "description": ""}, "v61": {"definition": "random(v60+1..125)", "templateType": "anything", "group": "Ungrouped variables", "name": "v61", "description": ""}, "v60": {"definition": "random(90..110)", "templateType": "anything", "group": "Ungrouped variables", "name": "v60", "description": ""}, "ans12": {"definition": "(-b+(sqrt(b^2-(4*c))))/2", "templateType": "anything", "group": "Ungrouped variables", "name": "ans12", "description": ""}, "ans13": {"definition": "(-b-(sqrt(b^2-(4*c))))/2", "templateType": "anything", "group": "Ungrouped variables", "name": "ans13", "description": ""}, "v2": {"definition": "random(2..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "v2", "description": ""}, "ans11": {"definition": "an12/an11", "templateType": "anything", "group": "Ungrouped variables", "name": "ans11", "description": ""}, "v4": {"definition": "random(22..35)", "templateType": "anything", "group": "Ungrouped variables", "name": "v4", "description": ""}, "an1": {"definition": "n1[0]^n1[1]", "templateType": "anything", "group": "Ungrouped variables", "name": "an1", "description": ""}, "v7": {"definition": "shuffle(2..5)[0..2]", "templateType": "anything", "group": "Ungrouped variables", "name": "v7", "description": ""}, "n21": {"definition": "n2[0]^ans12", "templateType": "anything", "group": "Ungrouped variables", "name": "n21", "description": ""}, "c": {"definition": "n10[0]*n10[1]-(na1[0]^na1[1])", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "n10[0]+n10[1]", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "v1": {"definition": "random(45..75)", "templateType": "anything", "group": "Ungrouped variables", "name": "v1", "description": ""}, "n8": {"definition": "random(2..4)", "templateType": "anything", "group": "Ungrouped variables", "name": "n8", "description": ""}, "v3": {"definition": "random(2..5 except v2)", "templateType": "anything", "group": "Ungrouped variables", "name": "v3", "description": ""}, "n1": {"definition": "shuffle(2..5)[0..5]", "templateType": "anything", "group": "Ungrouped variables", "name": "n1", "description": ""}, "n2": {"definition": "shuffle(3..7)[0..2]", "templateType": "anything", "group": "Ungrouped variables", "name": "n2", "description": ""}, "n7": {"definition": "random(2..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "n7", "description": ""}, "n4": {"definition": "shuffle(2..5)[0..2]", "templateType": "anything", "group": "Ungrouped variables", "name": "n4", "description": ""}, "n5": {"definition": "random(2..4)", "templateType": "anything", "group": "Ungrouped variables", "name": "n5", "description": ""}, "n6": {"definition": "random(4..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "n6", "description": ""}, "v5": {"definition": "shuffle(2..8)[0..4]", "templateType": "anything", "group": "Ungrouped variables", "name": "v5", "description": ""}}, "metadata": {"notes": "<p>rebelmaths rebel Rebel REBEL</p>", "description": "<p>Using Logs to solve equations involving indices</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Solving equations that contain natural log", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}], "functions": {}, "ungrouped_variables": ["n1", "n2", "n3", "ans1", "n4", "n5", "n6", "n7", "ans2", "n8", "n9", "n10", "ans3"], "tags": ["rebel", "rebelmaths"], "preamble": {"css": "", "js": ""}, "advice": "<p>i) $\\var{n1}\\ln(\\var{n2}x) = \\var{n3}$</p>\n<p>$\\ln(\\var{n2}x) = \\frac{\\var{n3}}{\\var{n1}}$</p>\n<p>Next,</p>\n<p>$\\var{n2}x = e^{\\frac{\\var{n3}}{\\var{n1}}}$</p>\n<p>$x= \\frac{e^{(\\frac{\\var{n3}}{\\var{n1}})} }{\\var{n2}} = \\var{ans1}$</p>\n<p></p>\n<p>ii) $\\var{n4}\\ln(\\frac{\\var{n5}x}{\\var{n6}}) = \\var{n7}$</p>\n<p>$\\ln(\\frac{\\var{n5}x}{\\var{n6}}) =\\frac{ \\var{n7}}{\\var{n4}}$</p>\n<p>Next,</p>\n<p>$\\frac{\\var{n5}x}{\\var{n6}}=e^{\\frac{ \\var{n7}}{\\var{n4}}}$</p>\n<p>$x= e^{(\\frac{\\var{n7}}{\\var{n4}})} \\times \\frac{\\var{n6}}{\\var{n5}}&nbsp;= \\var{ans2}$</p>\n<p></p>\n<p></p>\n<p>iii) $\\var{n8} = \\ln(\\frac{\\var{n9}}{\\var{n10}x})$</p>\n<p>$e^{\\var{n8}}=\\frac{\\var{n9}}{\\var{n10}x}$</p>\n<p>$\\var{n10}xe^{\\var{n8}}=\\var{n9}$</p>\n<p>$x= \\frac{\\var{n9}}{(\\var{n10}&nbsp;\\times e^\\var{n8})} = \\var{ans3}$</p>", "rulesets": {}, "parts": [{"prompt": "<p>i) $\\var{n1}\\ln(\\var{n2}x) = \\var{n3}$</p>\n<p>$x = $[[0]]</p>\n<p>ii) $\\var{n4}\\ln(\\frac{\\var{n5}x}{\\var{n6}}) = \\var{n7}$</p>\n<p>$x = $[[1]]</p>\n<p>iii) $\\var{n8} = \\ln(\\frac{\\var{n9}}{\\var{n10}x})$</p>\n<p><em>Give correct to 4 decimal places:</em></p>\n<p>$x = $[[2]]</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "precision": "2", "maxValue": "{ans1}", "variableReplacementStrategy": "originalfirst", "strictPrecision": false, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": 0, "scripts": {}, "marks": 1, "minValue": "{ans1}", "type": "numberentry", "showPrecisionHint": false}, {"precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "precision": "2", "maxValue": "{ans2}", "variableReplacementStrategy": "originalfirst", "strictPrecision": false, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": 0, "scripts": {}, "marks": 1, "minValue": "{ans2}", "type": "numberentry", "showPrecisionHint": false}, {"precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "precision": "4", "maxValue": "{ans3}", "variableReplacementStrategy": "originalfirst", "strictPrecision": false, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": 0, "scripts": {}, "marks": 1, "minValue": "{ans3}", "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "statement": "<p>Solve for $x$ in the following, correct to 2 decimal places:&nbsp;</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"n10": {"definition": "random(2..4)", "templateType": "anything", "group": "Ungrouped variables", "name": "n10", "description": ""}, "ans1": {"definition": "e^(n3/n1)/n2", "templateType": "anything", "group": "Ungrouped variables", "name": "ans1", "description": ""}, "ans2": {"definition": "e^(n7/n4)*(n6/n5)", "templateType": "anything", "group": "Ungrouped variables", "name": "ans2", "description": ""}, "ans3": {"definition": "n9/(n10*(e^n8))", "templateType": "anything", "group": "Ungrouped variables", "name": "ans3", "description": ""}, "n8": {"definition": "random(5.5..8.5#0.1 except [7,8])", "templateType": "anything", "group": "Ungrouped variables", "name": "n8", "description": ""}, "n9": {"definition": "random(3.5..5.4#0.1 except [4,5])", "templateType": "anything", "group": "Ungrouped variables", "name": "n9", "description": ""}, "n1": {"definition": "random(7..11)", "templateType": "anything", "group": "Ungrouped variables", "name": "n1", "description": ""}, "n2": {"definition": "random(n1+1..21)", "templateType": "anything", "group": "Ungrouped variables", "name": "n2", "description": ""}, "n3": {"definition": "random(n2..40)", "templateType": "anything", "group": "Ungrouped variables", "name": "n3", "description": ""}, "n4": {"definition": "random(7.1..8.9#0.1 except 8)", "templateType": "anything", "group": "Ungrouped variables", "name": "n4", "description": ""}, "n5": {"definition": "random(2.5..5.5#0.1)", "templateType": "anything", "group": "Ungrouped variables", "name": "n5", "description": ""}, "n6": {"definition": "random(6..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "n6", "description": ""}, "n7": {"definition": "random(5.1..6.9#0.1 except 6)", "templateType": "anything", "group": "Ungrouped variables", "name": "n7", "description": ""}}, "metadata": {"notes": "<p>rebelmaths rebel Rebel REBEL</p>", "description": "<p>Using e to solve equations involving the natural log</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}]}], "contributors": [{"name": "Deirdre Casey", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/681/"}], "extensions": [], "custom_part_types": [], "resources": [["question-resources/rules_73.jpg", "/srv/numbas/media/question-resources/rules_73.jpg"], ["question-resources/rules_73.jpg", "/srv/numbas/media/question-resources/rules_73.jpg"]]}