// Numbas version: exam_results_page_options
{"name": "Solve for x in equation with subtraction 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 subtraction 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, rounding your answer to $\\var{decimal_places}$ decimal place(s):</p>\n<p>\\[ \\log_{\\var{base}}(\\var{a}x+\\var{b})&nbsp;- \\log_{\\var{base}}(\\simplify{{c}x+{n*d}}) =&nbsp;\\var{f} \\]</p>", "advice": "<p>Beginning with the equation given:</p>\n<p></p>\n<p>\\[ \\log_{\\var{base}}(\\var{a}x+\\var{b})&nbsp;-&nbsp;\\log_{\\var{base}}(\\simplify{{c}x+{n*d}}) =&nbsp;\\var{f} ,\\]</p>\n<p></p>\n<p>note the following rule of logs that allows us to convert between the <strong>log of a quotient</strong> and <strong>the subtraction of two separate log terms</strong>:</p>\n<p></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 \\left( \\frac{x}{y} \\right) &amp; = \\log_a x - \\log_a y\\\\ \\\\ \\text{e.g., } \\log_3 \\left( \\frac{x}{9} \\right) &amp; = \\log_3 (x) - \\log_3 (9) \\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 subtracting.</p>\n<p style=\"text-align: left;\"></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;\">$\\begin{align*} \\underbrace{\\color{black}{\\log_{\\var{base}}(\\var{a}x+\\var{b}) - \\log_{\\var{base}}(\\simplify{{c}x+{n*d}})}} &amp;= \\var{f}\\\\ \\searrow \\\\ &amp; \\log_{\\var{base}}(\\var{a}x+\\var{b}) - \\log_{\\var{base}}(\\simplify{{c}x+{n*d}}) =\\log_{\\var{base}} \\left(\\frac{\\var{a}x+\\var{b}}{\\simplify{{c}x+{n*d}}}\\right) \\end{align*}$&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}}\\left(\\frac{\\var{a}x+\\var{b}}{\\simplify{{c}x+{n*d}}}\\right)= \\var{f} \\]</p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\">Next, note the following rule of logs that allows us to cancel a base and a log to the same base:</p>\n<p></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*} a^{\\log_a (x)} &amp; = x\\\\ \\\\ \\text{e.g., } 3^{\\log_3 (7)} &amp; = 7 \\end{align*}&nbsp; \\]</td>\n</tr>\n</tbody>\n</table>\n<p style=\"text-align: center;\"></p>\n<p style=\"text-align: left;\">Therefore, to cancel out the $\\textbf{LOG}$ to base $\\var{base}$, we must take both sides as a $\\textbf{POWER}$ on $\\var{base}$:</p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\">\\[ \\begin{align*}\\log_{\\var{base}} \\left( \\frac{\\var{a}x+\\var{b}}{\\simplify{{c}x+{n*d}}}\\right) &amp; = \\var{f}\\\\ \\phantom{.}^{\\var{base}^{(\\ \\ \\ )}} \\bigg\\downarrow&nbsp; &amp;&nbsp; \\hspace{0.5cm}&nbsp; \\bigg\\downarrow \\phantom{.}^{\\var{base}^{(\\ \\ \\ )}} \\\\ \\var{base}^{\\log_{\\var{base}} \\left( \\frac{\\var{a}x+\\var{b}}{\\simplify{{c}x+{n*d}}} \\right)} &amp; = \\var{base}^{\\var{f}} \\end{align*}\\]</p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\">We can now use rule of logs mentioned above to cancel the $\\log_{\\var{base}}$ with the base $\\var{base}$ on the left-hand side:</p>\n<p style=\"text-align: left;\">\\[ \\begin{align*} \\var{base}^{\\log_{\\var{base}} \\left( \\frac{\\var{a}x+\\var{b}}{\\simplify{{c}x+{n*d}}}\\right)} &amp; = \\var{base}^{\\var{f}} \\\\ \\frac{\\var{a}x+\\var{b}}{\\simplify{{c}x+{n*d}}} &amp; = \\var{base}^{\\var{f}} \\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;\">\\[&nbsp;\\frac{\\var{a}x+\\var{b}}{\\simplify{{c}x+{n*d}}} = \\simplify{{base}^{f}} \\]</p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\">We can now get rid of the fraction by $\\textbf{MULTIPLYING}$ both sides by the denominator $\\simplify{{c}x+{n*d}}$:</p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\">\\[ \\begin{align*} \\frac{\\var{a}x+\\var{b}}{\\simplify{{c}x+{n*d}}} \\cdot (\\simplify{{c}x+{n*d}}) &amp; =&nbsp; \\simplify{{base}^{f}} \\cdot (\\simplify{{c}x+{n*d}})\\\\ \\var{a}x+\\var{b} &amp; = \\simplify{{base}^{f}} \\cdot (\\simplify{{c}x+{n*d}}) \\end{align*} \\]</p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\">Now let's $\\textbf{MULTIPLY OUT}$ the brackets on the right-hand side:</p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\">\\[ \\var{a}x+\\var{b} = \\simplify{{base}^{f}*{c}*x+{base}^{f}*{n*d}} \\]</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 style=\"text-align: left;\">\\[ \\begin{align*} \\simplify[!noLeadingMinus,unitFactor]{{a}*x-{{base}^{f}*{c}}*x} &amp;= \\simplify[!noLeadingMinus]{{{base}^{f}*{n*d}}-{b}}\\\\ \\simplify[unitFactor]{{a-base^f*c}*x} &amp; = \\simplify{{base^f*n*d-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; = \\frac{\\var{numerator}}{\\var{denominator}}\\\\ &amp; \\hspace{-1cm}\\phantom{.}^{\\text{Calculator work}} \\bigg\\downarrow\\\\ x &amp;= \\var{ans} \\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..5)", "description": "<p>Base of log terms in the equation.</p>", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2..6)", "description": "<p>Multiple for x in the first log term.</p>", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(1..8)", "description": "<p>Integer to be added to x in the first log term.</p>", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(2..9 except a)", "description": "<p>Multiple for x in the second 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 d.</p>", "templateType": "anything", "can_override": false}, "x": {"name": "x", "group": "Ungrouped variables", "definition": "(g*n*d-b)/(a-g*c)", "description": "<p>Variable the student must solve for.</p>", "templateType": "anything", "can_override": false}, "decimal_places": {"name": "decimal_places", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "<p>Number of decimal places the student must round their answer to.</p>", "templateType": "anything", "can_override": false}, "numerator": {"name": "numerator", "group": "Ungrouped variables", "definition": "g*n*d-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-g*c", "description": "<p>Denominator of fraction in the second last line of the solution.</p>", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(1..5 except b)", "description": "<p>Integer to be added to or subtracted from x in the second log term.</p>", "templateType": "anything", "can_override": false}, "f": {"name": "f", "group": "Ungrouped variables", "definition": "random(2..4)", "description": "<p>Difference between the two log terms in the equation.</p>", "templateType": "anything", "can_override": false}, "g": {"name": "g", "group": "Ungrouped variables", "definition": "base^f", "description": "", "templateType": "anything", "can_override": false}, "ans": {"name": "ans", "group": "Ungrouped variables", "definition": "precround(x,decimal_places)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "a*x+b>0 and c*x+n*d>0", "maxRuns": 100}, "ungrouped_variables": ["base", "a", "b", "c", "d", "f", "g", "n", "decimal_places", "x", "numerator", "denominator", "ans"], "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": 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><span style=\"text-decoration: underline;\"><strong>Don't forget to round your answer&nbsp;to $\\var{decimal_places}$ decimal place(s).</strong></span></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": "{ans}", "maxValue": "{ans}", "correctAnswerFraction": false, "allowFractions": false, "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 - 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 equation given:</p>\n<p></p>\n<p>\\[ \\log_{\\var{base}}(\\var{a}x+\\var{b})&nbsp;-&nbsp;\\log_{\\var{base}}(\\simplify{{c}x+{n*d}}) =&nbsp;\\var{f} ,\\]</p>\n<p></p>\n<p>note the following rule of logs that allows us to convert between the <strong>log of a quotient</strong> and <strong>the subtraction of two separate log terms</strong>:</p>\n<p></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 \\left( \\frac{x}{y} \\right) &amp; = \\log_a x - \\log_a y\\\\ \\\\ \\text{e.g., } \\log_3 \\left( \\frac{x}{9} \\right) &amp; = \\log_3 (x) - \\log_3 (9) \\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 subtracting.</p>\n<p style=\"text-align: left;\"></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;\">$\\begin{align*} \\underbrace{\\color{black}{\\log_{\\var{base}}(\\var{a}x+\\var{b}) - \\log_{\\var{base}}(\\simplify{{c}x+{n*d}})}} &amp;= \\var{f}\\\\ \\searrow \\\\ &amp; \\log_{\\var{base}}(\\var{a}x+\\var{b}) - \\log_{\\var{base}}(\\simplify{{c}x+{n*d}}) =\\log_{\\var{base}} \\left(\\frac{\\var{a}x+\\var{b}}{\\simplify{{c}x+{n*d}}}\\right) \\end{align*}$&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}}\\left(\\frac{\\var{a}x+\\var{b}}{\\simplify{{c}x+{n*d}}}\\right)= \\var{f} \\]</p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\">Can you think about how you might use another rule of logs to tidy up the left-hand side a little more? To continue on with the question yourself, click \"Go back to the previous part\" below or select \"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 - finishing the question", "rawLabel": "", "otherPart": 3, "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;\">\\[&nbsp;\\log_{\\var{base}}\\left(\\frac{\\var{a}x+\\var{b}}{\\simplify{{c}x+{n*d}}}\\right)=&nbsp;\\var{f}&nbsp;\\]</p>\n<p style=\"text-align: left;\">Next, note the following rule of logs that allows us to cancel a base and a log to the same base:</p>\n<p></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*} a^{\\log_a (x)} &amp; = x\\\\ \\\\ \\text{e.g., } 3^{\\log_3 (7)} &amp; = 7 \\end{align*}&nbsp; \\]</td>\n</tr>\n</tbody>\n</table>\n<p style=\"text-align: center;\"></p>\n<p style=\"text-align: left;\">Therefore, to cancel out the $\\textbf{LOG}$ to base $\\var{base}$, we must take both sides as a $\\textbf{POWER}$ on $\\var{base}$:</p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\">\\[ \\begin{align*}\\log_{\\var{base}} \\left( \\frac{\\var{a}x+\\var{b}}{\\simplify{{c}x+{n*d}}}\\right) &amp; = \\var{f}\\\\ \\phantom{.}^{\\var{base}^{(\\ \\ \\ )}} \\bigg\\downarrow&nbsp; &amp;&nbsp; \\hspace{0.5cm}&nbsp; \\bigg\\downarrow \\phantom{.}^{\\var{base}^{(\\ \\ \\ )}} \\\\ \\var{base}^{\\log_{\\var{base}} \\left( \\frac{\\var{a}x+\\var{b}}{\\simplify{{c}x+{n*d}}} \\right)} &amp; = \\var{base}^{\\var{f}} \\end{align*}\\]</p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\">We can now use rule of logs mentioned above to cancel the $\\log_{\\var{base}}$ with the base $\\var{base}$ on the left-hand side:</p>\n<p style=\"text-align: left;\">\\[ \\begin{align*} \\var{base}^{\\log_{\\var{base}} \\left( \\frac{\\var{a}x+\\var{b}}{\\simplify{{c}x+{n*d}}}\\right)} &amp; = \\var{base}^{\\var{f}} \\\\ \\frac{\\var{a}x+\\var{b}}{\\simplify{{c}x+{n*d}}} &amp; = \\var{base}^{\\var{f}} \\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;\">\\[&nbsp;\\frac{\\var{a}x+\\var{b}}{\\simplify{{c}x+{n*d}}} = \\simplify{{base}^{f}} \\]</p>\n<p style=\"text-align: left;\">We now have a much more straightforward equation that no longer involves logs. 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 3 - finishing the question\" below.</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 style=\"text-align: left;\">Our equation now looks like:</p>\n<p style=\"text-align: left;\">\\[&nbsp;\\frac{\\var{a}x+\\var{b}}{\\simplify{{c}x+{n*d}}} = \\simplify{{base}^{f}} \\]</p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\">We can now get rid of the fraction by $\\textbf{MULTIPLYING}$ both sides by the denominator $\\simplify{{c}x+{n*d}}$:</p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\">\\[ \\begin{align*} \\frac{\\var{a}x+\\var{b}}{\\simplify{{c}x+{n*d}}} \\cdot (\\simplify{{c}x+{n*d}}) &amp; =&nbsp; \\simplify{{base}^{f}} \\cdot (\\simplify{{c}x+{n*d}})\\\\ \\var{a}x+\\var{b} &amp; = \\simplify{{base}^{f}} \\cdot (\\simplify{{c}x+{n*d}}) \\end{align*} \\]</p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\">Now let's $\\textbf{MULTIPLY OUT}$ the brackets on the right-hand side:</p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\">\\[ \\var{a}x+\\var{b} = \\simplify{{base}^{f}*{c}*x+{base}^{f}*{n*d}} \\]</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 style=\"text-align: left;\">\\[ \\begin{align*} \\simplify[!noLeadingMinus,unitFactor]{{a}*x-{{base}^{f}*{c}}*x} &amp;= \\simplify[!noLeadingMinus]{{{base}^{f}*{n*d}}-{b}}\\\\ \\simplify[unitFactor]{{a-base^f*c}*x} &amp; = \\simplify{{base^f*n*d-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; = \\frac{\\var{numerator}}{\\var{denominator}}\\\\ &amp; \\hspace{-1cm}\\phantom{.}^{\\text{Calculator work}} \\bigg\\downarrow\\\\ x &amp;= \\var{ans} \\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/"}]}