// Numbas version: exam_results_page_options
{"name": "Solve for x as a power", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Solve for x as a power", "tags": [], "metadata": {"description": "<p>Given an equation with variable x in the power on two different bases, students 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, rounding your answer to $\\var{dec}$ decimal places:</p>\n<p></p>\n<p>\\[\\var{a}^{\\simplify{{c}x+{d}}} = \\var{b}^{\\simplify{{f}x+{g}}}\\]</p>", "advice": "<p>Beginning with the equation we are given $\\var{a}^{\\simplify{{c}x+{d}}} = \\var{b}^{\\simplify{{f}x+{g}}}$,</p>\n<p></p>\n<p style=\"text-align: left;\">note the following rule of logs:</p>\n<table style=\"border-style: solid; border-color: #000000; margin-left: auto; margin-right: auto;\">\n<tbody>\n<tr>\n<td>\\[ \\hspace{0.5cm} \\begin{align*} \\log_a (x^p) &amp;= p \\cdot \\log _a (x)\\\\&nbsp; \\\\ \\text{e.g., } \\log_3 (7^4) &amp; = 4 \\cdot \\log_3(7) \\end{align*} \\hspace{0.5cm}\\]</td>\n</tr>\n</tbody>\n</table>\n<p style=\"text-align: left;\"></p>\n<p>Therefore, if we introduce logs, we can use this rule to get rid of the power of $\\simplify{{c}x+{d}}$ on the left and the power of $\\simplify{{f}x+{g}}$ on the right!&nbsp;</p>\n<p></p>\n<p>Since all calculators have a $\\log_{10}$ button (usually just written as $\\log$ on the calculator), we will use $\\log_{10}$ to do this:</p>\n<p></p>\n<p>\\[ \\begin{align*} \\var{a}^{\\simplify{{c}x+{d}}} &amp; = \\var{b}^{\\simplify{{f}x+{g}}}\\\\ \\log_{10} ( \\var{a}^{\\simplify{{c}x+{d}}} ) &amp; = \\log_{10} ( \\var{b}^{\\simplify{{f}x+{g}}} ) \\hspace{2cm} \\{\\textbf{Take } \\log_{10} \\textbf{ of both sides.} \\} \\end{align*} \\]</p>\n<p></p>\n<p>Now use the rule of logs mentioned in the box above to bring the powers out from the log terms:</p>\n<p></p>\n<p>\\[ \\begin{align*} \\log_{10} \\left( \\var{a}^{\\simplify{{c}x+{d}}} \\right) &amp; = \\log_{10} \\left(\\var{b}^{\\simplify{{f}x+{g}}} \\right)\\\\ \\\\ (\\simplify{{c}x+{d}}) \\log_{10} \\var{a} &amp; = (\\simplify{{f}x+{g}}) \\log_{10} \\var{b} \\end{align*} \\]</p>\n<p></p>\n<p>Next, $\\textbf{MULTIPLY OUT}$ the brackets on both sides:</p>\n<p></p>\n<p>\\[ \\begin{align*} (\\simplify{{c}x+{d}}) \\log_{10} \\var{a} &amp; = (\\simplify{{f}x+{g}}) \\log_{10} \\var{b}\\\\ \\\\ \\simplify[!unitFactor]{{c}x} \\log_{10} \\var{a} \\ \\ \\var{d_plus_or_minus} \\ \\ \\simplify{{abs_d}} \\log_{10} \\var{a} &amp; = \\simplify{{f}x} \\log_{10} \\var{b}\\ \\ \\var{g_plus_or_minus} \\ \\ \\var{abs_g} \\log_{10} \\var{b}&nbsp; \\hspace{2cm}\\{ \\textbf{Multiply out the brackets}\\}\\end{align*} \\]</p>\n<p style=\"text-align: left;\"></p>\n<table style=\"border-style: solid; border-color: #000000; margin-left: auto; margin-right: auto; height: 72px;\">\n<tbody>\n<tr style=\"height: 18px;\">\n<td style=\"text-align: center; vertical-align: middle; height: 18px; width: 335.097px;\"><strong>PLAN:&nbsp;</strong></td>\n</tr>\n<tr style=\"height: 18px;\">\n<td style=\"height: 18px; width: 335.097px; text-align: center; vertical-align: middle;\">Group all <strong>$x$-terms</strong> on one side</td>\n</tr>\n<tr style=\"height: 18px;\">\n<td style=\"height: 18px; width: 335.097px; text-align: center; vertical-align: middle;\">and all <strong>number terms </strong>on the other side.</td>\n</tr>\n<tr style=\"height: 18px;\">\n<td style=\"height: 18px; width: 335.097px; text-align: center; vertical-align: middle;\"><strong>BE VERY CAREFUL WITH THE SIGNS HERE!</strong></td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p></p>\n<p></p>\n<p>So let's make sure all terms that have an $x$ are on the left-hand side of the equation and all terms without an $x$ are on the right-hand side.</p>\n<p></p>\n<p>To cancel the $\\textbf{ADDITION}$ of $\\simplify{{f}x} \\log_{10} \\var{b}$ on the right-hand side, we must $\\textbf{SUBTRACT }$ $\\simplify{{f}x} \\log_{10} \\var{b}$ from both sides:</p>\n<p></p>\n<p>\\[ \\begin{align*} \\simplify[!unitFactor]{{c}x} \\log_{10} \\var{a} \\ \\ \\var{d_plus_or_minus} \\ \\ \\simplify{{abs_d}} \\log_{10} \\var{a} &amp; = \\simplify{{f}x} \\log_{10} \\var{b}\\ \\ \\var{g_plus_or_minus} \\ \\ \\var{abs_g} \\log_{10} \\var{b} \\\\ \\\\ \\hspace{-6cm} -\\simplify{{f}x} \\log_{10} \\var{b} \\bigg\\downarrow \\phantom{\\hspace{2.5cm}} &amp; \\hspace{0.75cm} \\bigg\\downarrow -\\simplify{{f}x} \\log_{10} \\var{b}\\\\ \\\\ \\simplify[!unitFactor]{{c}x} \\log_{10} \\var{a} \\ \\ \\var{d_plus_or_minus} \\ \\ \\simplify{{abs_d}} \\log_{10} \\var{a} - \\simplify{{f}x} \\log_{10} \\var{b} &amp; = \\var{g} \\log_{10} \\var{b} \\end{align*}\\]</p>\n<p></p>\n<p>To cancel the $\\var{addition_or_subtraction}$ of $\\simplify{(-1)^({n}) {d}} \\log_{10} \\var{a}$ on the left-hand side, we must $\\var{add_or_subtract}$ $\\simplify{(-1)^({n}) {d}} \\log_{10} \\var{a}$ $\\var{from_or_to}$ both sides:</p>\n<p></p>\n<p>\\[ \\begin{align*} \\simplify[!unitFactor]{{c}x} \\log_{10} \\var{a} \\ \\ \\var{d_plus_or_minus} \\ \\ \\simplify{{abs_d}} \\log_{10} \\var{a} - \\simplify{{f}x} \\log_{10} \\var{b} &amp; = \\var{g} \\log_{10} \\var{b} \\\\ \\\\ \\hspace{-6cm} \\var{not_d_plus_or_minus} \\ \\simplify{(-1)^({n}) {d}} \\log_{10} \\var{a} \\bigg\\downarrow \\phantom{\\hspace{2.5cm}} &amp; \\hspace{0.75cm} \\bigg\\downarrow \\var{not_d_plus_or_minus} \\ \\simplify{(-1)^({n}) {d}} \\log_{10} \\var{a}\\\\ \\\\ \\simplify[!unitFactor]{{c}x} \\log_{10} \\var{a}&nbsp; - \\simplify{{f}x} \\log_{10} \\var{b} &amp; = \\var{g} \\log_{10} \\var{b} \\ \\ \\var{not_d_plus_or_minus} \\ \\ \\simplify{(-1)^({n}) {d}} \\log_{10} \\var{a} \\end{align*}\\]</p>\n<p></p>\n<p>Now, note that both terms on the left-hand side have an $x$ in common:</p>\n<p>\\[ \\begin{align*} [\\var{c} \\log_{10} \\var{a}] x - [\\var{f} \\log_{10} \\var{b}] x &amp; = \\var{g} \\log_{10} \\var{b} \\ \\ \\var{not_d_plus_or_minus} \\ \\ \\simplify{(-1)^({n}) {d}} \\log_{10} \\var{a}\\\\ \\\\ \\bigg\\downarrow \\hspace{0.75cm} \\textbf{Factor out } &amp; \\textbf{the common }&nbsp; x \\textbf{ term} \\\\ \\\\ [\\var{c} \\log_{10} \\var{a} - \\var{f} \\log_{10} \\var{b}] x &amp; = \\var{g} \\log_{10} \\var{b} \\ \\ \\var{not_d_plus_or_minus} \\ \\ \\simplify{(-1)^({n}) {d}} \\log_{10} \\var{a} \\end{align*} \\]</p>\n<p>To cancel the $\\textbf{MULTIPLICATION}$ of $\\var{c} \\log_{10} \\var{a} - \\var{f} \\log_{10} \\var{b}$ on the left-hand side, we must $\\textbf{DIVIDE}$ both sides by $\\var{c} \\log_{10} \\var{a} - \\var{f} \\log_{10} \\var{b}$:</p>\n<p></p>\n<p>\\[ \\begin{align*} [\\var{c} \\log_{10} \\var{a} - \\var{f} \\log_{10} \\var{b}] x &amp; = \\var{g} \\log_{10} \\var{b} \\ \\ \\var{not_d_plus_or_minus} \\ \\ \\simplify{(-1)^({n}) {d}} \\log_{10} \\var{a} \\\\ \\\\ \\div \\ \\big[ \\var{c} \\log_{10} \\var{a} - \\var{f} \\log_{10} \\var{b} \\big] \\bigg\\downarrow \\phantom{\\hspace{2.5cm}} &amp; \\hspace{0.75cm} \\bigg\\downarrow \\div \\ \\big[ \\var{c} \\log_{10} \\var{a} - \\var{f} \\log_{10} \\var{b} \\big]\\\\ \\\\ x &amp; = \\frac{\\var{g} \\log_{10} \\var{b} \\ \\ \\var{not_d_plus_or_minus} \\ \\ \\simplify{(-1)^({n}) {d}} \\log_{10} \\var{a}}{\\var{c} \\log_{10} \\var{a} - \\var{f} \\log_{10} \\var{b}} \\\\ \\\\&nbsp; &amp; \\hspace{0.75cm} \\bigg\\downarrow \\{ \\textbf{Calculator work} \\}\\\\ \\\\ x &amp; =\\var{x}\\\\ \\\\ &amp; \\hspace{0.75cm} \\bigg\\downarrow \\textbf{Round to } \\var{dec} \\textbf{ decimal places} \\\\ \\\\ x &amp; = \\var{ans}&nbsp; \\end{align*} \\]</p>\n<p></p>\n<p>$\\textbf{Want some more practice? Click the \"Try another question like this one\" button at the end of this question.}$</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2..8)", "description": "<p>Base of number on left-hand side of equation.</p>", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(2..9 except a)", "description": "<p>Base of number on right-hand side of equation.</p>", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(2..8)", "description": "<p>Multiple of x in the power on the left-hand side.</p>", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(-6..6 except 0 except c)", "description": "<p>Number added to multiple of x in the power on the left-hand side.</p>", "templateType": "anything", "can_override": false}, "f": {"name": "f", "group": "Ungrouped variables", "definition": "random(1..9 except c)", "description": "<p>Multiple of x in the power on the right-hand side.</p>", "templateType": "anything", "can_override": false}, "g": {"name": "g", "group": "Ungrouped variables", "definition": "random(-3..6 except 0 except d except 1 except -1)", "description": "<p>Number added to multiple of x in the power on the right-hand side.</p>", "templateType": "anything", "can_override": false}, "dec": {"name": "dec", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "<p>Number of decimal places the student must round the answer to.</p>", "templateType": "anything", "can_override": false}, "x": {"name": "x", "group": "Ungrouped variables", "definition": "(g*log(b)-d*log(a))/(c*log(a)-f*log(b))", "description": "<p>Variable the student must solve for.</p>", "templateType": "anything", "can_override": false}, "ans": {"name": "ans", "group": "Ungrouped variables", "definition": "precround(x,dec)", "description": "<p>Answer rounded to correct number (\"dec\") of decimal places.</p>", "templateType": "anything", "can_override": false}, "addition_or_subtraction_vector": {"name": "addition_or_subtraction_vector", "group": "Ungrouped variables", "definition": "[\"ADDITION\",\"SUBTRACTION\"]", "description": "<p>Vector of strings/phrases required in the Advice and Hints sections.</p>", "templateType": "anything", "can_override": false}, "add_or_subtract_vector": {"name": "add_or_subtract_vector", "group": "Ungrouped variables", "definition": "[\"ADD\",\"SUBTRACT\"]", "description": "<p>Vector of strings/phrases required in the Advice and Hints sections.</p>", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "if(d>0,0,1)", "description": "<p>Used to determine which of the strings \"ADDITION\" and \"SUBTRACT\"; \"SUBTRACTION\" and \"ADD\"; and \"to\" and \"from\" are required in the solution given in the Advice and Hints sections.</p>\n<p>Determined by the variable d.</p>", "templateType": "anything", "can_override": false}, "addition_or_subtraction": {"name": "addition_or_subtraction", "group": "Ungrouped variables", "definition": "addition_or_subtraction_vector[n]", "description": "<p>Selecting the string \"ADDITION\" or \"SUBTRACTION\" from the variable addition_or_subtraction_vector, depending on whether the variable d has been added to or subtracted from the x term in the power on the left-hand side of the original equation.&nbsp;</p>\n<p>Used in the Advice and Hints sections.</p>", "templateType": "anything", "can_override": false}, "add_or_subtract": {"name": "add_or_subtract", "group": "Ungrouped variables", "definition": "add_or_subtract_vector[1-n]", "description": "<p>Selecting the string \"SUBTRACT\" or \"ADD\" from the variable add_or_subtract_vector, depending on whether the variable d has been added to or subtracted from the x term in the power on the left-hand side of the original equation.&nbsp;</p>\n<p>Used in the Advice and Hints sections.</p>", "templateType": "anything", "can_override": false}, "from_or_to_vector": {"name": "from_or_to_vector", "group": "Ungrouped variables", "definition": "[\"to\",\"from\"]", "description": "<p>Vector of strings required in the Advice/Hints sections.</p>", "templateType": "anything", "can_override": false}, "from_or_to": {"name": "from_or_to", "group": "Ungrouped variables", "definition": "from_or_to_vector[1-n]", "description": "<p>Selecting the string \"to\" or \"from\" from the variable from_or_to_vector, depending on whether the variable d has been added to or subtracted from the x term in the power on the left-hand side of the original equation.&nbsp;</p>\n<p>Used in the Advice and Hints sections.</p>", "templateType": "anything", "can_override": false}, "d_plus_or_minus_vector": {"name": "d_plus_or_minus_vector", "group": "Ungrouped variables", "definition": "[\"+\",\"-\"]", "description": "<p>Vector of operations required in Advice/Hints section, depending on whether the variable d was added to or subtracted from the x term in the power on the left-hand side of the original equation.</p>\n<p>Required to simplify expressions involving log_10 expressions where the \\simplify command would not work.</p>", "templateType": "anything", "can_override": false}, "d_plus_or_minus": {"name": "d_plus_or_minus", "group": "Ungrouped variables", "definition": "d_plus_or_minus_vector[n]", "description": "<p>Selecting the operator \"+\" or \"-\" from the variable d_plus_or_minus_vector, depending on whether the variable d has been added to or subtracted from the x term in the power on the left-hand side of the original equation.&nbsp;</p>\n<p>Used in the Advice and Hints sections.</p>", "templateType": "anything", "can_override": false}, "abs_d": {"name": "abs_d", "group": "Ungrouped variables", "definition": "abs(d)", "description": "<p>Absolute value of the variable d, required in the Advice/Hints sections to simplify expressions involving log_10 expressions where the \\simplify command would not work.</p>", "templateType": "anything", "can_override": false}, "g_plus_or_minus_vector": {"name": "g_plus_or_minus_vector", "group": "Ungrouped variables", "definition": "[\"+\",\"-\"]", "description": "<p>Vector of operations required in Advice/Hints section, depending on whether the variable g was added to or subtracted from the x term in the power on the right-hand side of the original equation.</p>\n<p>Required to simplify expressions involving log_10 expressions where the \\simplify command would not work.</p>", "templateType": "anything", "can_override": false}, "g_plus_or_minus": {"name": "g_plus_or_minus", "group": "Ungrouped variables", "definition": "g_plus_or_minus_vector[m]", "description": "<p>Selecting the operator \"+\" or \"-\" from the variable g_plus_or_minus_vector, depending on whether the variable g has been added to or subtracted from the x term in the power on the right-hand side of the original equation.&nbsp;</p>\n<p>Used in the Advice and Hints sections.</p>", "templateType": "anything", "can_override": false}, "abs_g": {"name": "abs_g", "group": "Ungrouped variables", "definition": "abs(g)", "description": "<p>Absolute value of the variable g, required in the Advice/Hints sections to simplify expressions involving log_10 expressions where the \\simplify command would not work.</p>", "templateType": "anything", "can_override": false}, "not_d_plus_or_minus": {"name": "not_d_plus_or_minus", "group": "Ungrouped variables", "definition": "d_plus_or_minus_vector[1-n]", "description": "<p>Selecting the operator \"+\" or \"-\" from the variable d_plus_or_minus_vector, depending on whether the variable d has been added to or subtracted from the x term in the power on the left-hand side of the original equation.&nbsp;</p>\n<p>This is the complement of the variable d_plus_or_minus.</p>\n<p>Used in the Advice and Hints sections.</p>", "templateType": "anything", "can_override": false}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "if(g>0,0,1)", "description": "<p>Used to determine which of \"+\" or \"-\" are required in the solution given in the Advice and Hints sections.</p>\n<p>Determined by the variable g.</p>", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "c", "d", "f", "g", "dec", "x", "ans", "addition_or_subtraction_vector", "add_or_subtract_vector", "n", "addition_or_subtraction", "add_or_subtract", "from_or_to_vector", "from_or_to", "d_plus_or_minus_vector", "d_plus_or_minus", "abs_d", "g_plus_or_minus_vector", "m", "g_plus_or_minus", "abs_g", "not_d_plus_or_minus"], "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 - some more help", "rawLabel": "", "otherPart": 3, "variableReplacements": [], "availabilityCondition": "", "penalty": "", "penaltyAmount": 0, "showPenaltyHint": true, "lockAfterLeaving": false}, {"label": "Hint 4 - some more help", "rawLabel": "", "otherPart": 4, "variableReplacements": [], "availabilityCondition": "", "penalty": "", "penaltyAmount": 0, "showPenaltyHint": true, "lockAfterLeaving": false}, {"label": "Hint 5 - finishing the question", "rawLabel": "", "otherPart": 5, "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": "", "showFractionHint": 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 - some more help", "rawLabel": "", "otherPart": 3, "variableReplacements": [], "availabilityCondition": "", "penalty": "", "penaltyAmount": 0, "showPenaltyHint": true, "lockAfterLeaving": false}, {"label": "Hint 4 - some more help", "rawLabel": "", "otherPart": 4, "variableReplacements": [], "availabilityCondition": "", "penalty": "", "penaltyAmount": 0, "showPenaltyHint": true, "lockAfterLeaving": false}, {"label": "Hint 5 - finishing the question", "rawLabel": "", "otherPart": 5, "variableReplacements": [], "availabilityCondition": "", "penalty": "", "penaltyAmount": 0, "showPenaltyHint": true, "lockAfterLeaving": false}], "suggestGoingBack": true, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Beginning with the equation we are given $\\var{a}^{\\simplify{{c}x+{d}}} = \\var{b}^{\\simplify{{f}x+{g}}}$,</p>\n<p></p>\n<p style=\"text-align: left;\">note the following rule of logs:</p>\n<table style=\"border-style: solid; border-color: #000000; margin-left: auto; margin-right: auto;\">\n<tbody>\n<tr>\n<td>\\[ \\hspace{0.5cm} \\begin{align*} \\log_a (x^p) &amp;= p \\cdot \\log _a (x)\\\\&nbsp; \\\\ \\text{e.g., } \\log_3 (7^4) &amp; = 4 \\cdot \\log_3(7) \\end{align*} \\hspace{0.5cm}\\]</td>\n</tr>\n</tbody>\n</table>\n<p style=\"text-align: left;\"></p>\n<p>Therefore, if we introduce logs, we can use this rule to get rid of the power of $\\simplify{{c}x+{d}}$ on the left and the power of $\\simplify{{f}x+{g}}$ on the right!&nbsp;</p>\n<p></p>\n<p>Since all calculators have a $\\log_{10}$ button (usually just written as $\\log$ on the calculator), we will use $\\log_{10}$ to do this:</p>\n<p></p>\n<p>\\[ \\begin{align*} \\var{a}^{\\simplify{{c}x+{d}}} &amp; = \\var{b}^{\\simplify{{f}x+{g}}}\\\\ \\log_{10} ( \\var{a}^{\\simplify{{c}x+{d}}} ) &amp; = \\log_{10} ( \\var{b}^{\\simplify{{f}x+{g}}} ) \\hspace{2cm} \\{\\textbf{Take } \\log_{10} \\textbf{ of both sides.} \\} \\end{align*} \\]</p>\n<p></p>\n<p>Now see if you can use the rule of logs mentioned in the box above to bring the powers out from the log terms.</p>\n<p>To continue with the question yourself, click \"Go back to the previous part\" below or click \"Solve for $x$\" in the progress bar at the top of the question. Otherwise, if you need some more help, click \"Hint 2 - a little more help\" for a futher 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 - some more help", "rawLabel": "", "otherPart": 3, "variableReplacements": [], "availabilityCondition": "", "penalty": "", "penaltyAmount": 0, "showPenaltyHint": true, "lockAfterLeaving": false}, {"label": "Hint 4 - some more help", "rawLabel": "", "otherPart": 4, "variableReplacements": [], "availabilityCondition": "", "penalty": "", "penaltyAmount": 0, "showPenaltyHint": true, "lockAfterLeaving": false}, {"label": "Hint 5 - finishing the question", "rawLabel": "", "otherPart": 5, "variableReplacements": [], "availabilityCondition": "", "penalty": "", "penaltyAmount": 0, "showPenaltyHint": true, "lockAfterLeaving": false}], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Our equation now looks like:</p>\n<p></p>\n<p>\\[ \\log_{10} ( \\var{a}^{\\simplify{{c}x+{d}}} ) = \\log_{10} ( \\var{b}^{\\simplify{{f}x+{g}}} ) \\]</p>\n<p></p>\n<p>We can now use the rule of logs mentioned earlier to bring the powers out from the log terms:</p>\n<p></p>\n<p>\\[ \\begin{align*} \\log_{10} \\left( \\var{a}^{\\simplify{{c}x+{d}}} \\right) &amp; = \\log_{10} \\left(\\var{b}^{\\simplify{{f}x+{g}}} \\right)\\\\ \\\\ (\\simplify{{c}x+{d}}) \\log_{10} \\var{a} &amp; = (\\simplify{{f}x+{g}}) \\log_{10} \\var{b} \\end{align*} \\]</p>\n<p></p>\n<p>Next, try $\\textbf{MULTIPLYING OUT}$ the brackets on both sides (without evaluating the log terms) and see if you can continue from there.</p>\n<p></p>\n<p>To continue with the question yourself, click \"Solve for $x$\" in the progress bar at the top of the question. Otherwise, click \"Hint 3 - some more help\" if you would like a further hint.</p>"}, {"type": "information", "useCustomName": true, "customName": "Hint 3 - some 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 2 - a little more help", "rawLabel": "", "otherPart": 2, "variableReplacements": [], "availabilityCondition": "", "penalty": "", "penaltyAmount": 0, "showPenaltyHint": true, "lockAfterLeaving": false}, {"label": "Hint 4 - some more help", "rawLabel": "", "otherPart": 4, "variableReplacements": [], "availabilityCondition": "", "penalty": "", "penaltyAmount": 0, "showPenaltyHint": true, "lockAfterLeaving": false}, {"label": "Hint 5 - finishing the question", "rawLabel": "", "otherPart": 5, "variableReplacements": [], "availabilityCondition": "", "penalty": "", "penaltyAmount": 0, "showPenaltyHint": true, "lockAfterLeaving": false}], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Starting with our simpler equation</p>\n<p>\\[&nbsp;(\\simplify{{c}x+{d}}) \\log_{10}(\\var{a}) = (\\simplify{{f}x+{g}}) \\log_{10}(\\var{b}) \\]</p>\n<p>we can now $\\textbf{MULTIPLY OUT}$ the brackets on both sides:</p>\n<p></p>\n<p>\\[ \\begin{align*} (\\simplify{{c}x+{d}}) \\log_{10} \\var{a} &amp; = (\\simplify{{f}x+{g}}) \\log_{10} \\var{b}\\\\ \\\\ \\Rightarrow \\simplify[!unitFactor]{{c}x} \\log_{10} \\var{a} \\ \\ \\var{d_plus_or_minus} \\ \\ \\simplify{{abs_d}} \\log_{10} \\var{a} &amp; = \\simplify{{f}x} \\log_{10} \\var{b}\\ \\ \\var{g_plus_or_minus} \\ \\ \\var{abs_g} \\log_{10} \\var{b}\\end{align*} \\]</p>\n<p style=\"text-align: left;\"></p>\n<table style=\"border-style: solid; border-color: #000000; margin-left: auto; margin-right: auto; height: 72px;\">\n<tbody>\n<tr style=\"height: 18px;\">\n<td style=\"text-align: center; vertical-align: middle; height: 18px; width: 335.097px;\"><strong>PLAN:&nbsp;</strong></td>\n</tr>\n<tr style=\"height: 18px;\">\n<td style=\"height: 18px; width: 335.097px; text-align: center; vertical-align: middle;\">Group all <strong>$x$-terms</strong> on one side</td>\n</tr>\n<tr style=\"height: 18px;\">\n<td style=\"height: 18px; width: 335.097px; text-align: center; vertical-align: middle;\">and all <strong>number terms </strong>on the other side.</td>\n</tr>\n<tr style=\"height: 18px;\">\n<td style=\"height: 18px; width: 335.097px; text-align: center; vertical-align: middle;\"><strong>BE VERY CAREFUL WITH THE SIGNS HERE!</strong></td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p>Can you use this plan to get a little further in the question?</p>\n<p>To continue with the question yourself, click \"Solve for $x$\" in the progress bar at the top of the question. Otherwise, click \"Hint 4 - some more help\" if you would like another hint.</p>"}, {"type": "information", "useCustomName": true, "customName": "Hint 4 - some 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 2 - a little more help", "rawLabel": "", "otherPart": 2, "variableReplacements": [], "availabilityCondition": "", "penalty": "", "penaltyAmount": 0, "showPenaltyHint": true, "lockAfterLeaving": false}, {"label": "Hint 3 - some more help", "rawLabel": "", "otherPart": 3, "variableReplacements": [], "availabilityCondition": "", "penalty": "", "penaltyAmount": 0, "showPenaltyHint": true, "lockAfterLeaving": false}, {"label": "Hint 5 - finishing the question", "rawLabel": "", "otherPart": 5, "variableReplacements": [], "availabilityCondition": "", "penalty": "", "penaltyAmount": 0, "showPenaltyHint": true, "lockAfterLeaving": false}], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>We now wish to group all of the $x \\text{ terms}$ together and group all of the $\\text{number terms}$ together in our equation:</p>\n<p>\\[ \\simplify[!unitFactor]{{c}x} \\log_{10} \\var{a} \\ \\ \\var{d_plus_or_minus} \\ \\ \\simplify{{abs_d}} \\log_{10} \\var{a} = \\simplify{{f}x} \\log_{10} \\var{b}\\ \\ \\var{g_plus_or_minus} \\ \\ \\var{abs_g} \\log_{10} \\var{b} \\]</p>\n<p></p>\n<p>To cancel the $\\textbf{ADDITION}$ of $\\simplify{{f}x} \\log_{10} \\var{b}$ on the right-hand side, we must $\\textbf{SUBTRACT }$ $\\simplify{{f}x} \\log_{10} \\var{b}$ from both sides:</p>\n<p></p>\n<p>\\[ \\begin{align*} \\simplify[!unitFactor]{{c}x} \\log_{10} \\var{a} \\ \\ \\var{d_plus_or_minus} \\ \\ \\simplify{{abs_d}} \\log_{10} \\var{a} &amp; = \\simplify{{f}x} \\log_{10} \\var{b}\\ \\ \\var{g_plus_or_minus} \\ \\ \\var{abs_g} \\log_{10} \\var{b} \\\\ \\\\ \\hspace{-6cm} -\\simplify{{f}x} \\log_{10} \\var{b} \\bigg\\downarrow \\phantom{\\hspace{2.5cm}} &amp; \\hspace{0.75cm} \\bigg\\downarrow -\\simplify{{f}x} \\log_{10} \\var{b}\\\\ \\\\ \\simplify[!unitFactor]{{c}x} \\log_{10} \\var{a} \\ \\ \\var{d_plus_or_minus} \\ \\ \\simplify{{abs_d}} \\log_{10} \\var{a} - \\simplify{{f}x} \\log_{10} \\var{b} &amp; = \\var{g} \\log_{10} \\var{b} \\end{align*}\\]</p>\n<p></p>\n<p>To cancel the $\\var{addition_or_subtraction}$ of $\\simplify{(-1)^({n}) {d}} \\log_{10} \\var{a}$ on the left-hand side, we must $\\var{add_or_subtract}$ $\\simplify{(-1)^({n}) {d}} \\log_{10} \\var{a}$ $\\var{from_or_to}$ both sides:</p>\n<p></p>\n<p>\\[ \\begin{align*} \\simplify[!unitFactor]{{c}x} \\log_{10} \\var{a} \\ \\ \\var{d_plus_or_minus} \\ \\ \\simplify{{abs_d}} \\log_{10} \\var{a} - \\simplify{{f}x} \\log_{10} \\var{b} &amp; = \\var{g} \\log_{10} \\var{b} \\\\ \\\\ \\hspace{-6cm} \\var{not_d_plus_or_minus} \\ \\simplify{(-1)^({n}) {d}} \\log_{10} \\var{a} \\bigg\\downarrow \\phantom{\\hspace{2.5cm}} &amp; \\hspace{0.75cm} \\bigg\\downarrow \\var{not_d_plus_or_minus} \\ \\simplify{(-1)^({n}) {d}} \\log_{10} \\var{a}\\\\ \\\\ \\simplify[!unitFactor]{{c}x} \\log_{10} \\var{a}&nbsp; - \\simplify{{f}x} \\log_{10} \\var{b} &amp; = \\var{g} \\log_{10} \\var{b} \\ \\ \\var{not_d_plus_or_minus} \\ \\ \\simplify{(-1)^({n}) {d}} \\log_{10} \\var{a} \\end{align*}\\]</p>\n<p></p>\n<p>Now, note that both terms on the left-hand side have an $x$ in common:</p>\n<p>\\[ \\begin{align*} [\\var{c} \\log_{10} \\var{a}] x - [\\var{f} \\log_{10} \\var{b}] x &amp; = \\var{g} \\log_{10} \\var{b} \\ \\ \\var{not_d_plus_or_minus} \\ \\ \\simplify{(-1)^({n}) {d}} \\log_{10} \\var{a}\\\\ \\\\ \\bigg\\downarrow \\hspace{0.75cm} \\textbf{Factor out } &amp; \\textbf{the common }&nbsp; x \\textbf{ term} \\\\ \\\\ [\\var{c} \\log_{10} \\var{a} - \\var{f} \\log_{10} \\var{b}] x &amp; = \\var{g} \\log_{10} \\var{b} \\ \\ \\var{not_d_plus_or_minus} \\ \\ \\simplify{(-1)^({n}) {d}} \\log_{10} \\var{a} \\end{align*} \\]</p>\n<p></p>\n<p>Can you finish the question from here?&nbsp;</p>\n<p>To continue with the question yourself, click \"Solve for $x$\" in the progress bar at the top of the question. Otherwise, click \"Hint 5 - finishing the question\" to see how to go through the final steps.</p>"}, {"type": "information", "useCustomName": true, "customName": "Hint 5 - 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>To finish the question, we must now get $x$ on its own. This means we must&nbsp;cancel the $\\textbf{MULTIPLICATION}$ of $\\var{c} \\log_{10} \\var{a} - \\var{f} \\log_{10} \\var{b}$ on the left-hand side. We do this by $\\textbf{DIVIDING}$ both sides by $\\var{c} \\log_{10} \\var{a} - \\var{f} \\log_{10} \\var{b}$:</p>\n<p></p>\n<p>\\[ \\begin{align*} [\\var{c} \\log_{10} \\var{a} - \\var{f} \\log_{10} \\var{b}] x &amp; = \\var{g} \\log_{10} \\var{b} \\ \\ \\var{not_d_plus_or_minus} \\ \\ \\simplify{(-1)^({n}) {d}} \\log_{10} \\var{a} \\\\ \\\\ \\div \\ \\big[ \\var{c} \\log_{10} \\var{a} - \\var{f} \\log_{10} \\var{b} \\big] \\bigg\\downarrow \\phantom{\\hspace{1.75cm}} &amp; \\hspace{2.5cm} \\bigg\\downarrow \\div \\ \\big[ \\var{c} \\log_{10} \\var{a} - \\var{f} \\log_{10} \\var{b} \\big]\\\\ \\\\ x &amp; = \\frac{\\var{g} \\log_{10} \\var{b} \\ \\ \\var{not_d_plus_or_minus} \\ \\ \\simplify{(-1)^({n}) {d}} \\log_{10} \\var{a}}{\\var{c} \\log_{10} \\var{a} - \\var{f} \\log_{10} \\var{b}} \\\\ \\\\&nbsp; &amp; \\hspace{0.75cm} \\bigg\\downarrow \\{ \\textbf{Calculator work} \\}\\\\ \\\\ x &amp; =\\var{x}\\\\ \\\\ &amp; \\hspace{0.75cm} \\bigg\\downarrow \\textbf{Round to } \\var{dec} \\textbf{ decimal places} \\\\ \\\\ x &amp; = \\var{ans}&nbsp; \\end{align*} \\]</p>\n<p></p>\n<p>$\\textbf{Want some more practice? Click the \"Try another question like this one\" button at the end of this question.}$</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/"}]}