// Numbas version: exam_results_page_options
{"name": "Solve for x in an equation where x is a power on e", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Solve for x in an equation where x is a power on e", "tags": [], "metadata": {"description": "<p>Given an equation involving x as a power on e, the student must solve for x. Hints are included in the question to aid the student as needed.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Solve for $x$ in the following equation, giving your answer to 3 decimal places:</p>\n<p>\\[&nbsp; \\var{a}e^{\\var{b}x} = \\var{c}\\]</p>", "advice": "<p>Beginning with the given equation:</p>\n<p>\\[ \\var{a}e^{\\var{b}x} = \\var{c}, \\]</p>\n<p></p>\n<p>we first divide both sides by $\\var{a}$ - the number in front of the exponential term - so that the exponential term will then be on its own:</p>\n<p></p>\n<p></p>\n<p>\\[\\begin{align*} \\frac{\\var{a}e^{\\var{b}x}}{\\var{a}} &amp; = \\frac{\\var{c}}{\\var{a}} \\hspace{2cm} \\{\\textbf{Divide both sides by } \\var{a} \\}\\\\ \\textbf{DIVISION by } \\var{a}&nbsp; \\hspace{3cm} &amp;\\\\ \\textbf{cancels } \\hspace{1.5cm} \\swarrow \\phantom{\\hspace{1.5cm}} &amp; \\\\ \\textbf{MULTIPLICATION&nbsp; by }&nbsp; \\var{a} \\hspace{2.5cm} &amp;\\\\ \\searrow \\hspace{1.65cm} &amp;\\\\ e^{\\var{b}x} &amp; = \\simplify{{c}/{a}}\\end{align*}\\]</p>\n<p></p>\n<p>Next, note the following rule of logs that allows us to cancel a base and a log to the same base:</p>\n<table style=\"height: 36px; border-style: solid; border-color: #000000; margin-left: auto; margin-right: auto;\">\n<tbody>\n<tr style=\"height: 18px;\">\n<td style=\"text-align: center; height: 18px; width: 377px;\">\\[ \\begin{align*} \\log_a (a^x) &amp; = x\\\\ \\\\ \\text{e.g., } \\log_4 (4^{10}) &amp; = 10\\\\ \\\\ \\ln (e^{2x}) &amp;= \\log_e(e^{2x}) = 2x \\end{align*}&nbsp; \\]</td>\n</tr>\n</tbody>\n</table>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\">Therefore, we can take $\\ln$ (i.e., $\\log_e$) of both sides to get rid of the base $e$ on the left-hand side:</p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\">\\[\\begin{align*} \\log_e \\left( e^{\\var{b}x} \\right) &amp; = \\log_e \\left( \\simplify{{c}/{a}}\\right) \\hspace{2cm} \\{\\textbf{Take } \\log_e \\textbf{ of both sides.} \\}\\\\ \\textbf{LOG} \\textbf{ to the base } e \\hspace{0.5cm} \\swarrow \\hspace{1.5cm}&amp;\\\\ \\textbf{cancels the base } \\ e&nbsp; \\phantom{\\hspace{2.5cm}}&amp; \\\\ \\textbf{ INSIDE the brackets} \\hspace{2.2cm}\\\\ \\searrow \\hspace{1.65cm} \\\\ \\var{b}x &amp; = \\log_e \\left( \\simplify{{c}/{a}}\\right)\\\\ \\\\ \\text{Next, we can divide both sides by } \\var{b} \\text{ to get } x \\text{ on its own:}\\\\ \\\\ \\frac{\\var{b}x}{\\var{b}} &amp; = \\frac{\\log_e \\left( \\simplify{{c}/{a}}\\right)}{\\var{b}} \\hspace{2cm} \\{\\textbf{Divide both sides by } \\var{b} \\}\\\\ \\textbf{DIVISION} \\textbf{ by } \\var{b} \\hspace{3cm} &amp; \\\\ \\textbf{cancels } \\hspace{1.5cm}&nbsp; \\swarrow \\phantom{\\hspace{1.5cm}} &amp; \\\\ \\textbf{MULTIPLICATION} \\textbf{ by } \\ \\var{b} \\hspace{2.5cm}&amp;\\\\ \\searrow \\hspace{1.65cm}\\\\ x &amp; = \\frac{\\log_e \\left( \\simplify{{c}/{a}}\\right)}{\\var{b}}\\\\ \\\\ &amp; \\hspace{-1cm} \\phantom{.}^{\\text{Calculator work}} \\bigg\\downarrow\\\\ \\\\ x &amp; = \\var{x}\\\\ \\\\ x &amp; = \\var{ans} \\hspace{2cm} \\{\\textbf{Round to }&nbsp; 3 \\textbf{ decimal places} \\}&nbsp; \\end{align*}\\]</p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\">$\\textbf{Want some more practice? Click the \"Try another question like this one\" button at the end of the question.}$</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "<p>Multiple of e.</p>", "templateType": "anything", "can_override": false}, "b1": {"name": "b1", "group": "Ungrouped variables", "definition": "random(-9 .. 9#0.1)", "description": "<p>Used to generate the power on the exponential. The power will be equal to b1 unless b1=0, in which case the power will be b1+0.1 (given by variable b below).</p>", "templateType": "randrange", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(2..10 except a except 2a except 3a except 4a except 5a)", "description": "<p>Right-hand side of the equation the student must solve (integer).</p>", "templateType": "anything", "can_override": false}, "x": {"name": "x", "group": "Ungrouped variables", "definition": "1/b*ln(c/a)", "description": "<p>Variable the student must solve for.</p>", "templateType": "anything", "can_override": false}, "ans": {"name": "ans", "group": "Ungrouped variables", "definition": "precround(x,3)", "description": "<p>Answer rounded to 3 decimal places.</p>", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "if(b1=0,b1+0.1,b1)", "description": "<p>Power on the exponential.</p>", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "gcd(a,c)=1", "maxRuns": 100}, "ungrouped_variables": ["a", "b1", "c", "x", "ans", "b"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": true, "customName": "Solve for $x$", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [{"label": "Hint 1 - how to start", "rawLabel": "", "otherPart": 1, "variableReplacements": [], "availabilityCondition": "", "penalty": "", "penaltyAmount": 0, "showPenaltyHint": true, "lockAfterLeaving": false}, {"label": "Hint 2 - a little more help", "rawLabel": "", "otherPart": 2, "variableReplacements": [], "availabilityCondition": "", "penalty": "", "penaltyAmount": 0, "showPenaltyHint": true, "lockAfterLeaving": false}, {"label": "Hint 3 - finishing the question", "rawLabel": "", "otherPart": 3, "variableReplacements": [], "availabilityCondition": "", "penalty": "", "penaltyAmount": 0, "showPenaltyHint": true, "lockAfterLeaving": false}], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": "Solve for $x$", "prompt": "<p>$x = $ [[0]]</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "studentNumber (The student's answer, parsed as a number):\n    if(settings[\"allowFractions\"],\n        parsedecimal_or_fraction(studentAnswer,settings[\"notationStyles\"])\n    ,\n        parsedecimal(studentAnswer,settings[\"notationStyles\"])\n    )\n\nisInteger (Is the student's answer an integer?):\n    countDP(studentAnswer)=0\n\nisFraction (Is the student's answer a fraction?):\n    \"/\" in studentAnswer\n\nnumerator (The numerator of the student's answer, or 0 if not a fraction):\n    if(isFraction,\n        parseNumber(split(studentAnswer,\"/\")[0],settings[\"notationStyles\"])\n    ,\n        0\n    )\n\ndenominator (The numerator of the student's answer, or 0 if not a fraction):\n    if(isFraction,\n        parseNumber(split(studentAnswer,\"/\")[1],settings[\"notationStyles\"])\n    ,\n        0\n    )\n\ncancelled (Is the student's answer a cancelled fraction?):\n    assert(isFraction and gcd(numerator,denominator)=1,\n        assert(not settings[\"mustBeReduced\"],\n            multiply_credit(settings[\"mustBeReducedPC\"],translate(\"part.numberentry.answer not reduced\"))\n        );\n        false\n    )\n\ncleanedStudentAnswer:\n    cleannumber(studentAnswer, settings[\"notationStyles\"])\n\nstudentPrecision:\n    switch(\n        settings[\"precisionType\"]=\"dp\", max(settings[\"precision\"],countDP(cleanedStudentAnswer)),\n        settings[\"precisionType\"]=\"sigfig\", max(settings[\"precision\"],countsigfigs(cleanedStudentAnswer)),\n        0\n    )\n\nraw_minvalue:\n    switch(\n        settings[\"precisionType\"]=\"dp\", precround(settings[\"minvalue\"],studentPrecision),\n        settings[\"precisionType\"]=\"sigfig\", siground(settings[\"minvalue\"],studentPrecision),\n        settings[\"minvalue\"]\n    )\n\nraw_maxvalue:\n    switch(\n        settings[\"precisionType\"]=\"dp\", precround(settings[\"maxvalue\"],studentPrecision),\n        settings[\"precisionType\"]=\"sigfig\", siground(settings[\"maxvalue\"],studentPrecision),\n        settings[\"maxvalue\"]\n    )\n\nminvalue: min(raw_minvalue,raw_maxvalue)\n\nmaxvalue: max(raw_minvalue,raw_maxvalue)\n\nvalidNumber (Is the student's answer a valid number?):\n    if(isNaN(studentNumber),\n        warn(translate(\"part.numberentry.answer invalid\"));\n        fail(translate(\"part.numberentry.answer invalid\"))\n    ,\n        true\n    )\n\nnumberInRange (Is the student's number in the allowed range?):\n    if(studentNumber>=minvalue and studentNumber<=maxvalue,\n        correct(\"Well done! For some more practice, click on the $``$Try another question like this one$\\\"$ button at the end of the question.\")\n    ,\n        incorrect(\"Not there yet. Have you looked at the hints provided below?\");\n        end()\n    )\n\ncorrectPrecision (Has the student's answer been given to the desired precision?):     \n    if(togivenprecision(cleanedStudentAnswer,settings['precisionType'],settings['precision'],settings[\"strictPrecision\"]),\n        true\n    ,\n        multiply_credit(settings[\"precisionPC\"],settings[\"precisionMessage\"]);\n        false\n    )\n\nmark (Mark the student's answer):\n    apply(validNumber);\n    apply(numberInRange);\n    assert(numberInRange,end());\n    if(isFraction,\n        apply(cancelled)\n    ,\n        apply(correctPrecision)\n    )\n \ninterpreted_answer (The student's answer, to be reused by other parts):\n    apply(validNumber);\n    studentNumber\n", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": "Solve for $x$", "minValue": "ans", "maxValue": "ans", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "precisionType": "dp", "precision": "3", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "information", "useCustomName": true, "customName": "Hint 1 - how to start", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [{"label": "Hint 2 - a little more help", "rawLabel": "", "otherPart": 2, "variableReplacements": [], "availabilityCondition": "", "penalty": "", "penaltyAmount": 0, "showPenaltyHint": true, "lockAfterLeaving": false}, {"label": "Hint 3 - finishing the question", "rawLabel": "", "otherPart": 3, "variableReplacements": [], "availabilityCondition": "", "penalty": "", "penaltyAmount": 0, "showPenaltyHint": true, "lockAfterLeaving": false}], "suggestGoingBack": true, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Beginning with the given equation:</p>\n<p>\\[ \\var{a}e^{\\var{b}x} = \\var{c}, \\]</p>\n<p></p>\n<p>we first divide both sides by $\\var{a}$ - the number in front of the exponential term - so that the exponential term will then be on its own:</p>\n<p></p>\n<p></p>\n<p>\\[\\begin{align*} \\frac{\\var{a}e^{\\var{b}x}}{\\var{a}} &amp; = \\frac{\\var{c}}{\\var{a}} \\hspace{2cm} \\{\\textbf{Divide both sides by } \\var{a} \\}\\\\ \\textbf{DIVISION by } \\var{a}&nbsp; \\hspace{3cm} &amp;\\\\ \\textbf{cancels } \\hspace{1.5cm} \\swarrow \\phantom{\\hspace{1.5cm}} &amp; \\\\ \\textbf{MULTIPLICATION&nbsp; by }&nbsp; \\var{a} \\hspace{2.5cm} &amp;\\\\ \\searrow \\hspace{1.65cm} &amp;\\\\ e^{\\var{b}x} &amp; = \\simplify{{c}/{a}}\\end{align*}\\]</p>\n<p></p>\n<p>Can you now think of a rule of logs that might help you to simplify the left-hand side and continue on with the question?</p>\n<p>To continue the question by yourself, click \"Go back to the previous part\" below or click \"Solve for $x$\" in the question progress bar at the top of the question. Otherwise, if you need some more help, click \"Hint 2 - a little more help\" below for a further hint.</p>"}, {"type": "information", "useCustomName": true, "customName": "Hint 2 - a little more help", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [{"label": "Hint 1 - how to start", "rawLabel": "", "otherPart": 1, "variableReplacements": [], "availabilityCondition": "", "penalty": "", "penaltyAmount": 0, "showPenaltyHint": true, "lockAfterLeaving": false}, {"label": "Hint 3 - finishing the question", "rawLabel": "", "otherPart": 3, "variableReplacements": [], "availabilityCondition": "", "penalty": "", "penaltyAmount": 0, "showPenaltyHint": true, "lockAfterLeaving": false}], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Our equation is now a little simpler than before. Since we now have</p>\n<p>\\[e^{\\var{b}x} = \\simplify{{c}/{a}}, \\]</p>\n<p>we can simplify the left-hand side by using the following rule of logs that allows us to cancel a base and a log to the same base:</p>\n<table style=\"height: 36px; border-style: solid; border-color: #000000; margin-left: auto; margin-right: auto;\">\n<tbody>\n<tr style=\"height: 18px;\">\n<td style=\"text-align: center; height: 18px; width: 377px;\">\\[ \\begin{align*} \\log_a (a^x) &amp; = x\\\\ \\\\ \\text{e.g., } \\log_4 (4^{10}) &amp; = 10\\\\ \\\\ \\ln (e^{2x}) &amp;= \\log_e(e^{2x}) = 2x \\end{align*}&nbsp; \\]</td>\n</tr>\n</tbody>\n</table>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\">Therefore, we can take $\\ln$ (i.e., $\\log_e$) of both sides to get rid of the base $e$ on the left-hand side:</p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\">\\[\\begin{align*} \\log_e \\left( e^{\\var{b}x} \\right) &amp; = \\log_e \\left( \\simplify{{c}/{a}}\\right) \\hspace{2cm} \\{\\textbf{Take } \\log_e \\textbf{ of both sides.} \\}\\\\ \\textbf{LOG} \\textbf{ to the base } e \\hspace{0.5cm} \\swarrow \\hspace{1.5cm}&amp;\\\\ \\textbf{cancels the base } \\ e&nbsp; \\phantom{\\hspace{2.5cm}}&amp; \\\\ \\textbf{ INSIDE the brackets} \\hspace{2.2cm}\\\\ \\searrow \\hspace{1.65cm} \\\\ \\var{b}x &amp; = \\log_e \\left( \\simplify{{c}/{a}}\\right) \\end{align*}\\]</p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\">Can you now find the value of $x$?</p>\n<p style=\"text-align: left;\">To continue with the question yourself, click \"Solve for $x$\" in the question progress bar at the top of the question. Otherwise, if you need more help, click \"Hint 3 - finishing the question\" below for help with the final steps.</p>"}, {"type": "information", "useCustomName": true, "customName": "Hint 3 - finishing the question", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": true, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Now our equation is</p>\n<p></p>\n<p>\\[ \\var{b}x = \\log_e \\left( \\simplify{{c}/{a}}\\right)\\]</p>\n<p></p>\n<p>To get $x$ on its own, we must get rid of the $\\var{b}$ that is being $\\textbf{MULTIPLIED}$ by $x$. We therefore $\\textbf{DIVIDE BOTH SIDES}$ by $\\var{b}$:</p>\n<p></p>\n<p>\\[\\begin{align*}&nbsp; \\frac{\\var{b}x}{\\var{b}} &amp; = \\frac{\\log_e \\left( \\simplify{{c}/{a}}\\right)}{\\var{b}} \\hspace{2cm} \\{\\textbf{Divide both sides by } \\var{b} \\}\\\\ \\textbf{DIVISION by } \\var{b} \\hspace{3cm} &amp; \\\\ \\textbf{cancels } \\hspace{1.5cm}&nbsp; \\swarrow \\phantom{\\hspace{1.5cm}} &amp; \\\\ \\textbf{MULTIPLICATION by } \\ \\var{b} \\hspace{2.5cm}&amp;\\\\ \\searrow \\hspace{1.65cm}\\\\ x &amp; = \\frac{\\log_e \\left( \\simplify{{c}/{a}}\\right)}{\\var{b}}\\\\ \\\\ &amp; \\hspace{-1cm}\\phantom{.}^{\\text{Calculator work}} \\bigg\\downarrow\\\\ \\\\ x &amp; = \\var{x}\\\\ \\\\ x &amp; = \\var{ans} \\hspace{2cm} \\{\\textbf{Round to } 3 \\textbf{ decimal places} \\}&nbsp; \\end{align*}\\]</p>\n<p></p>\n<p>$\\textbf{Want some more practice? Click the \"Try another question like this one\" button at the end of the question.}$</p>"}], "partsMode": "explore", "maxMarks": "2", "objectives": [{"name": "Solve for $x$", "limit": "2"}], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Clodagh Carroll", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/384/"}]}]}], "contributors": [{"name": "Clodagh Carroll", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/384/"}]}