// Numbas version: exam_results_page_options
{"name": "Solve for x in equation with addition of logs", "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 equation with addition of logs", "tags": [], "metadata": {"description": "<p>Given an equation with log terms added together, 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:</p>\n<p>\\[ \\log_{\\var{base}}(x+\\var{a}) + \\log_{\\var{base}}(\\simplify{x+{n*b}}) = 2 \\log_{\\var{base}}(\\simplify{x+{m*c}}) \\]</p>", "advice": "<p>Beginning with the equation given:</p>\n<p></p>\n<p>\\[ \\log_{\\var{base}}(x+\\var{a}) + \\log_{\\var{base}}(\\simplify{x+{n*b}}) = 2 \\log_{\\var{base}}(\\simplify{x+{m*c}}) ,\\]</p>\n<p></p>\n<p>note the following rule of logs that allows us to convert between the <strong>log of a product</strong> and <strong>the addition of two separate log terms</strong>:</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 (xy) &amp; = \\log_a x + \\log_a y\\\\ \\\\ \\text{e.g., } \\log_2 (7x) &amp; = \\log_2 (7) + \\log_2 (x) \\end{align*}&nbsp; \\]</td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p style=\"text-align: left;\">Remember, we can go from left to right or from right to left with this rule. The important thing is that the $\\textbf{BASE}$ is the same in the two log terms we are adding.</p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\">Applying this to the left-hand side of the equation above:</p>\n<p style=\"text-align: left;\">$\\begin{align*} \\underbrace{\\log_{\\var{base}}(x+\\var{a}) + \\log_{\\var{base}}(\\simplify{x+{n*b}})} &amp;= 2 \\log_{\\var{base}}(\\simplify{x+{m*c}})\\\\ \\searrow \\\\ &amp; \\log_{\\var{base}}(x+\\var{a}) + \\log_{\\var{base}}(\\simplify{x+{n*b}}) =\\log_{\\var{base}} [(x+\\var{a})(\\simplify{x+{n*b}})] \\end{align*}$&nbsp;</p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\">In other words,</p>\n<p style=\"text-align: left;\">\\[ \\log_{\\var{base}}[(x+\\var{a})(\\simplify{x+{n*b}})] = 2 \\log_{\\var{base}}(\\simplify{x+{m*c}}) \\]</p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\">Now, for the right-hand side, 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 style=\"text-align: left;\">Applying this to the right-hand side of our equation:</p>\n<p style=\"text-align: left;\">$\\begin{align*} \\log_{\\var{base}}[(x+\\var{a}) (\\simplify{x+{n*b}})] = \\underbrace{2 \\log_{\\var{base}}(\\simplify{x+{m*c}})} &amp;\\\\ \\searrow \\\\ &amp; 2 \\log_{\\var{base}}(\\simplify{x+{m*c}}) =\\log_{\\var{base}}[(\\simplify{x+{m*c}})^2] \\end{align*}$</p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\">In other words,</p>\n<p style=\"text-align: left;\">\\[ \\log_{\\var{base}}[(x+\\var{a}) (\\simplify{x+{n*b}})] = \\log_{\\var{base}}[(\\simplify{x+{m*c}})^2] \\]</p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\">We can now $\\textbf{\"CANCEL\"}$ the $\\log_{\\var{base}}$ on both sides and this gives:</p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\">\\[ (x+\\var{a}) (\\simplify{x+{n*b}}) = (\\simplify{x+{m*c}})^2 \\]</p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\">Next, $\\textbf{MULTIPLY OUT}$ the brackets on both sides:</p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\">\\[ \\begin{align*} \\underbrace{(x+ \\var{a}) (\\simplify{x+{n*b}})} &amp; = \\underbrace{(\\simplify{x+{m*c}})^2}\\\\ \\hspace{-2cm} \\bigg\\downarrow &amp; \\phantom{\\hspace{1.5cm}} \\searrow\\\\&nbsp; x(\\simplify{x + {n*b}}) +\\var{a}(\\simplify{x+{n*b}})&amp; = \\underbrace{(\\simplify{x+{m*c}})(\\simplify{x+{m*c}})} \\\\ \\hspace{-2.5cm} \\bigg\\downarrow &amp; \\phantom{\\hspace{1.8cm}}\\bigg\\downarrow\\\\ \\simplify{x^2 + {n*b}*x} +\\simplify{{a}x+{a*n*b}}&nbsp; &amp; =\\simplify[!noLeadingMinus]{x^2+{m*c}*x+{m*c}*x+{m*c}^2}\\\\ \\simplify{x^2+{a+n*b}*x+{a*n*b}} &amp; = \\simplify{x^2+{2*m*c}*x+{(m*c)^2}} \\end{align*} \\]</p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\">Next, we can subtract $x^2$ from both sides:</p>\n<p style=\"text-align: left;\">\\[\\begin{align*} \\simplify{x^2+{a+n*b}*x+{a*n*b}} -x^2 &amp; = \\simplify{x^2+{2*m*c}*x+{(m*c)^2}} -x^2\\\\ \\simplify{{a+n*b}*x+{a*n*b}} &amp; =&nbsp; \\simplify[!noLeadingMinus]{{2*m*c}*x+{(m*c)^2}} \\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 style=\"text-align: center;\"></p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\">\\[ \\begin{align*} \\simplify[!noLeadingMinus,unitFactor]{{a+n*b}*x-{2*m*c}*x} &amp;= \\simplify[!noLeadingMinus]{{(m*c)^2}-{a*n*b}}\\\\ \\simplify[unitFactor]{{a+n*b-2*m*c}*x} &amp; = \\simplify{{(m*c)^2-{a*n*b}}}\\\\ \\\\ \\text{Now, if there is a number in front of } &amp; x, \\text{ we can divide both sides by this number to get: }\\\\ \\\\ x &amp; = \\simplify{{numerator}/{denominator}}\\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>\n<p></p>\n<p style=\"text-align: left;\"></p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"base": {"name": "base", "group": "Ungrouped variables", "definition": "random(2..10)", "description": "<p>Base of log terms in the equation.</p>", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(1..6)", "description": "<p>Integer to be added to x in the first log term.</p>", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(1..8 except a)", "description": "<p>Integer to be added to or subtracted from x in the second log term.</p>", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(1..9 except 0.5*(a+n*b)/m)", "description": "<p>Integer to be added to or subtracted from x in the third log term.</p>", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(-1,1)", "description": "<p>Multiplier to assign random sign to the variable b.</p>", "templateType": "anything", "can_override": false}, "x": {"name": "x", "group": "Ungrouped variables", "definition": "(c^2-a*n*b)/(a+n*b-2m*c)", "description": "<p>Variable the student must solve for.</p>", "templateType": "anything", "can_override": false}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "random(-1,1)", "description": "<p>Multiplier to assign random sign to the variable c.</p>", "templateType": "anything", "can_override": false}, "numerator": {"name": "numerator", "group": "Ungrouped variables", "definition": "c^2-a*n*b", "description": "<p>Numerator of fraction in the second last line of the solution.</p>", "templateType": "anything", "can_override": false}, "denominator": {"name": "denominator", "group": "Ungrouped variables", "definition": "a+n*b-2m*c", "description": "<p>Denominator of fraction in the second last line of the solution.</p>", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "x+a>0 and x+n*b>0 and x+m*c>0", "maxRuns": 100}, "ungrouped_variables": ["base", "a", "b", "c", "n", "m", "x", "numerator", "denominator"], "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 - finishing the question", "rawLabel": "", "otherPart": 4, "variableReplacements": [], "availabilityCondition": "", "penalty": "", "penaltyAmount": 0, "showPenaltyHint": true, "lockAfterLeaving": false}], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>See if you can start the question by yourself first. If you need a hint, click on \"Hint 1 - how to start\" below.</p>\n<p>Write your answer as a whole number or as a fraction, using the <strong>/</strong> key. For example, to enter the fraction $-\\frac{2}{3}$, type&nbsp;<strong>$-2/3$</strong></p>\n<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": "(c^2-a*n*b)/(a+n*b-2m*c)", "maxValue": "(c^2-a*n*b)/(a+n*b-2m*c)", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": false, "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 - finishing the question", "rawLabel": "", "otherPart": 4, "variableReplacements": [], "availabilityCondition": "", "penalty": "", "penaltyAmount": 0, "showPenaltyHint": true, "lockAfterLeaving": false}], "suggestGoingBack": true, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Beginning with the equation given:</p>\n<p></p>\n<p>\\[ \\log_{\\var{base}}(x+\\var{a}) + \\log_{\\var{base}}(\\simplify{x+{n*b}}) = 2 \\log_{\\var{base}}(\\simplify{x+{m*c}}) ,\\]</p>\n<p></p>\n<p>note the following rule of logs that allows us to convert between the <strong>log of a product</strong> and <strong>the addition of two separate log terms</strong>:</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 (xy) &amp; = \\log_a x + \\log_a y\\\\ \\\\ \\text{e.g., } \\log_2 (7x) &amp; = \\log_2 (7) + \\log_2 (x) \\end{align*}&nbsp; \\]</td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p style=\"text-align: left;\">Remember, we can go from left to right or from right to left with this rule. The important thing is that the $\\textbf{BASE}$ is the same in the two log terms we are adding.</p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\">Applying this to the left-hand side of the equation above:</p>\n<p style=\"text-align: left;\">$\\begin{align*} \\underbrace{\\log_{\\var{base}}(x+\\var{a}) + \\log_{\\var{base}}(\\simplify{x+{n*b}})} &amp;= 2 \\log_{\\var{base}}(\\simplify{x+{m*c}})\\\\ \\searrow \\\\ &amp; \\log_{\\var{base}}(x+\\var{a}) + \\log_{\\var{base}}(\\simplify{x+{n*b}}) =\\log_{\\var{base}} [(x+\\var{a})(\\simplify{x+{n*b}})] \\end{align*}$&nbsp;</p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\">In other words,</p>\n<p style=\"text-align: left;\">\\[ \\log_{\\var{base}}[(x+\\var{a})(\\simplify{x+{n*b}})] = 2 \\log_{\\var{base}}(\\simplify{x+{m*c}}) \\]</p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\">Can you think of how to use another rule of logs to tidy up the right-hand side? To continue on with the question 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 with this part, click \"Hint 2 - a little more help\" below.</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 - finishing the question", "rawLabel": "", "otherPart": 4, "variableReplacements": [], "availabilityCondition": "", "penalty": "", "penaltyAmount": 0, "showPenaltyHint": true, "lockAfterLeaving": false}], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p style=\"text-align: left;\">We have now simplified the left-hand side a little bit and our equation now looks like:</p>\n<p style=\"text-align: left;\">\\[ \\log_{\\var{base}}[(x+\\var{a}) (\\simplify{x+{n*b}})] =&nbsp;2 \\log_{\\var{base}}(\\simplify{x+{m*c}}) \\]</p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\">Next, we take a look at the right-hand side. To simplify the right-hand side, 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 style=\"text-align: left;\">Applying this to the right-hand side of our equation:</p>\n<p style=\"text-align: left;\">$\\begin{align*} \\log_{\\var{base}}[(x+\\var{a}) (\\simplify{x+{n*b}})] = \\underbrace{2 \\log_{\\var{base}}(\\simplify{x+{m*c}})} &amp;\\\\ \\searrow \\\\ &amp; 2 \\log_{\\var{base}}(\\simplify{x+{m*c}}) =\\log_{\\var{base}}[(\\simplify{x+{m*c}})^2] \\end{align*}$</p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\">In other words,</p>\n<p style=\"text-align: left;\">\\[ \\log_{\\var{base}}[(x+\\var{a}) (\\simplify{x+{n*b}})] = \\log_{\\var{base}}[(\\simplify{x+{m*c}})^2] \\]</p>\n<p style=\"text-align: left;\">What is the next step? See if you can get a little further in the question. To continue on with the question yourself, click \"Solve for $x$\" in the question progress bar at the top of the question. Otherwise, if you need some more help, click \"Hint 3 - some more help\" below.</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 - finishing the question", "rawLabel": "", "otherPart": 4, "variableReplacements": [], "availabilityCondition": "", "penalty": "", "penaltyAmount": 0, "showPenaltyHint": true, "lockAfterLeaving": false}], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p style=\"text-align: left;\">Our equation now looks like:</p>\n<p style=\"text-align: left;\">\\[ \\log_{\\var{base}}[(x+\\var{a}) (\\simplify{x+{n*b}})] = \\log_{\\var{base}}[(\\simplify{x+{m*c}})^{2}] \\]</p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\">We can now $\\textbf{\"CANCEL\"}$ the $\\log_{\\var{base}}$ on both sides and this gives:</p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\">\\[ (x+\\var{a}) (\\simplify{x+{n*b}}) = (\\simplify{x+{m*c}})^2 \\]</p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\">We now have a much more straightforward equation that does not involve any $\\log$ terms. Can you now solve this equation for $x$? To continue on with the question yourself, click \"Solve for $x$\" in the question progress bar at the top of the question. Otherwise, if you need some help finishing the question, click \"Hint 4 - finishing the question\" below.</p>"}, {"type": "information", "useCustomName": true, "customName": "Hint 4 - 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 style=\"text-align: left;\">We now have a simpler equation than the one we started off with:</p>\n<p style=\"text-align: left;\">\\[ (x+\\var{a}) (\\simplify{x+{n*b}}) = (\\simplify{x+{m*c}})^{2} \\]</p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\">To solve this equation for $x$, let's $\\textbf{MULTIPLY OUT}$ the brackets on both sides:</p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\">\\[ \\begin{align*} \\underbrace{(x+ \\var{a}) (\\simplify{x+{n*b}})} &amp; = \\underbrace{(\\simplify{x+{m*c}})^2}\\\\ \\hspace{-2cm} \\bigg\\downarrow &amp; \\phantom{\\hspace{1.5cm}} \\searrow\\\\&nbsp; x(\\simplify{x + {n*b}}) +\\var{a}(\\simplify{x+{n*b}})&amp; = \\underbrace{(\\simplify{x+{m*c}})(\\simplify{x+{m*c}})} \\\\ \\hspace{-2.5cm} \\bigg\\downarrow &amp; \\phantom{\\hspace{1.8cm}}\\bigg\\downarrow\\\\ \\simplify{x^2 + {n*b}*x} +\\simplify{{a}x+{a*n*b}}&nbsp; &amp; =\\simplify[!noLeadingMinus]{x^2+{m*c}*x+{m*c}*x+{m*c}^2}\\\\ \\simplify{x^2+{a+n*b}*x+{a*n*b}} &amp; = \\simplify{x^2+{2*m*c}*x+{(m*c)^2}} \\end{align*} \\]</p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\">Next, we can subtract $x^2$ from both sides:</p>\n<p style=\"text-align: left;\">\\[\\begin{align*} \\simplify{x^2+{a+n*b}*x+{a*n*b}} -x^2 &amp; = \\simplify{x^2+{2*m*c}*x+{(m*c)^2}} -x^2\\\\ \\simplify{{a+n*b}*x+{a*n*b}} &amp; =&nbsp; \\simplify[!noLeadingMinus]{{2*m*c}*x+{(m*c)^2}} \\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 style=\"text-align: center;\"></p>\n<p style=\"text-align: left;\">\\[\\begin{align*} \\simplify[!noLeadingMinus,unitFactor]{{a+n*b}*x-{2*m*c}*x} &amp;= \\simplify[!noLeadingMinus]{{(m*c)^2}-{a*n*b}}\\\\ \\simplify[unitFactor]{{a+n*b-2*m*c}*x} &amp; = \\simplify{{(m*c)^2-{a*n*b}}}\\\\ \\\\ \\text{Now, if there is a number in front of } &amp; x, \\text{ we can divide both sides by this number to get: }\\\\ \\\\ x &amp; = \\simplify{{numerator}/{denominator}}\\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>"}], "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/"}]}