// Numbas version: exam_results_page_options
{"name": "David's copy of Logarithms: Solving equations 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "metadata": {"description": "<p>Solve for $x$: &nbsp;$\\log_{a}(x+b)- \\log_{a}(x+c)=d$</p>", "notes": "<p><strong>5/08/2012:</strong></p>\n<p>Added tags.</p>\n<p>Added description.</p>\n<p>Checked calculation.OK.</p>\n<p>Improved display in content areas.</p>\n<p>rebelmaths rebel Rebel REBEL</p>", "licence": "Creative Commons Attribution 4.0 International"}, "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  ", "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"description": "", "name": "a", "templateType": "anything", "definition": "random(2..5)", "group": "Ungrouped variables"}, "c": {"description": "", "name": "c", "templateType": "anything", "definition": "b-random(1..20)", "group": "Ungrouped variables"}, "b": {"description": "", "name": "b", "templateType": "anything", "definition": "random(1..20)", "group": "Ungrouped variables"}, "d": {"description": "", "name": "d", "templateType": "anything", "definition": "random(1,2)", "group": "Ungrouped variables"}}, "parts": [{"variableReplacements": [], "variableReplacementStrategy": "originalfirst", "stepsPenalty": 1, "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      ", "marks": 0, "gaps": [{"answersimplification": "std", "checkingaccuracy": 0.0001, "type": "jme", "vsetrangepoints": 5, "checkingtype": "absdiff", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "checkvariablenames": false, "notallowed": {"partialCredit": 0, "strings": ["."], "showStrings": false, "message": "<p>Input as a fraction or an integer, not as a decimal.</p>"}, "variableReplacements": [], "scripts": {}, "vsetrange": [0, 1], "showpreview": true, "marks": 2, "answer": "{b-c*a^d}/{a^d-1}", "expectedvariablenames": []}], "steps": [{"variableReplacements": [], "variableReplacementStrategy": "originalfirst", "prompt": "<p>Two rules for logs should be used:</p>\n<p>1.&nbsp;$\\log_a(\\frac{x}{y})=\\log_a(x)-\\log_a(y)$</p>\n<p>2.&nbsp;$a^x=y \\iff \\log_a y=x$</p>\n<p>Use rule 1 followed by rule 2 to get an equation for $x$.</p>", "marks": 0, "showCorrectAnswer": true, "scripts": {}, "type": "information"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}], "advice": "<p>We use the following two rules for logs :</p>\n<p>1.&nbsp;$\\log_a(\\frac{x}{y})=\\log_a(x)-\\log_a(y)$</p>\n<p>2.&nbsp;$a^x=y \\iff \\log_a y=x$</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>", "tags": ["algebra", "algebraic manipulation", "combining logarithms", "logarithm laws", "logarithms", "rebel", "rebelmaths", "simplifying logarithms", "solving equations", "Solving equations", "steps", "Steps"], "variable_groups": [], "preamble": {"js": "", "css": ""}, "showQuestionGroupNames": false, "name": "David's copy of Logarithms: Solving equations 1", "ungrouped_variables": ["a", "c", "b", "d"], "question_groups": [{"pickQuestions": 0, "questions": [], "name": "", "pickingStrategy": "all-ordered"}], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "type": "question", "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}, {"name": "David Martin", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/130/"}]}]}], "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}, {"name": "David Martin", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/130/"}]}