// Numbas version: exam_results_page_options
{"name": "Solving equations", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "name": "Solving equations", "tags": ["Steps", "algebra", "algebraic manipulation", "combining logarithms", "logarithm laws", "logarithms", "simplifying logarithms", "solving equations", "steps"], "advice": "\n  <p>We use the following two rules for logs :</p>\n  <p>1. $\\log_a(b)-\\log_a(c)=\\log_a(b/c)$</p>\n  <p>2. $\\log_a(p)=r \\Rightarrow p=a^r$</p>\n  <p>Using rule 1 we get<br/>\\[\\log_{\\var{a}}(x+\\var{b})- \\log_{\\var{a}}(\\simplify{(x+{c})})=\\log_{\\var{a}}\\left(\\simplify{(x+{b})/(x+{c})}\\right)\\]<br/>So the equation to solve becomes:<br/>\\[\\log_{\\var{a}}\\left(\\simplify{(x+{b})/(x+{c})}\\right)=\\var{d}\\]<br/>and using rule 2 this gives:<br/>\\[ \\begin{eqnarray} \\simplify{(x+{b})/(x+{c})}&amp;=&amp;{\\var{a}}^{\\var{d}}\\Rightarrow\\\\ x+\\var{b}&amp;=&amp;{\\var{a}}^{\\var{d}}(x+\\var{c})=\\simplify{{a^d}}(x+\\var{c})\\Rightarrow\\\\ \\simplify{{a^d-1}x}&amp;=&amp;\\simplify[std]{{b}-{c}*{a^d}={b-c*a^d}}\\Rightarrow\\\\ x&amp;=&amp;\\simplify{{b-c*a^d}/{a^d-1}} \\end{eqnarray} \\]<br/>We should check that this solution gives positive values for $x+\\var{b}$ and $\\simplify{x+{c}}$ as otherwise the logs are not defined.</p>\n  <p>Substituting this value for $x$ into $\\log_{\\var{a}}(x+\\var{b})$ we get $\\log_{\\var{a}}(\\simplify{({b-c }{a^d})/{a^d-1}})$ so OK.</p>\n  <p>For $\\log_{\\var{a}}(\\simplify{x+{c}})$ we get on substituting for $x$, $\\log_{\\var{a}}(\\simplify{({b-c })/{a^d-1}})$ so OK.</p>\n  <p>Hence the value we found for $x$ is a solution to the original equation.</p>\n  ", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "parts": [{"stepspenalty": 1.0, "prompt": "\n      <p>\\[\\log_{\\var{a}}(x+\\var{b})- \\log_{\\var{a}}(\\simplify{(x+{c})})=\\var{d}\\]</p>\n      <p>$x=\\;$ [[0]]</p>\n      <p>If you want help in solving the equation, click on Show steps. If you do so then you will lose 1 mark.</p>\n      <p>Input all numbers as fractions or integers and not as decimals.</p>\n      ", "gaps": [{"notallowed": {"message": "<p>Input as a fraction or an integer, not as a decimal.</p>", "showstrings": false, "strings": ["."], "partialcredit": 0.0}, "checkingaccuracy": 0.0001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 2.0, "answer": "{b-c*a^d}/{a^d-1}", "type": "jme"}], "steps": [{"prompt": "\n          <p>Two rules for logs should be used:</p>\n          <p>1. $\\log_a(b)-\\log_a(c)=\\log_a(b/c)$</p>\n          <p>2. $\\log_a(p)=r \\Rightarrow p=a^r$</p>\n          <p>Use rule 1 followed by rule 2 to get an equation for $x$.</p>\n          ", "type": "information", "marks": 0.0}], "marks": 0.0, "type": "gapfill"}], "extensions": [], "statement": "\n  <p>Solve the following equation for $x$.</p>\n  <p>Input your answer as a fraction or an integer as appropriate and not&nbsp;as a decimal.</p>\n  ", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "random(2..5)", "name": "a"}, "c": {"definition": "b-random(1..20)", "name": "c"}, "b": {"definition": "random(1..20)", "name": "b"}, "d": {"definition": "random(1,2)", "name": "d"}}, "metadata": {"notes": "\n    \t\t<p><strong>5/08/2012:</strong></p>\n    \t\t                <p>Added tags.</p>\n    \t\t                <p>Added description.</p>\n    \t\t                <p>Checked calculation.OK.</p>\n    \t\t                <p>Improved display in content areas.</p>\n    \t\t", "description": "<p>Solve for $x$: &nbsp;$\\log_{a}(x+b)- \\log_{a}(x+c)=d$</p>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}