// Numbas version: finer_feedback_settings
{"name": "Logarithms: Solving equations 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "name": "Logarithms: Solving equations 2", "tags": ["Steps", "algebra", "algebraic manipulation", "combining logarithms", "logarithm laws", "logarithms", "simplifying logarithms", "solving", "solving equations", "steps"], "advice": "\n  <p>We use the following three rules for logs :</p>\n  <p>1. $n\\log_a(m)=\\log_a(m^n)$</p>\n  <p>2. $\\log_a(b)-\\log_a(c)=\\log_a(b/c)$</p>\n  <p>3. $\\log_a(p)=r \\Rightarrow p=a^r$</p>\n  <p>Using rule 1 we get<br/>\\[2\\log_{\\var{a}}(\\simplify{x+{b}})- \\log_{\\var{a}}(\\simplify{(x+{c})})=\\log_{\\var{a}}((\\simplify{x+{b}})^2)- \\log_{\\var{a}}(\\simplify{(x+{c})})\\]<br/>Using rule 2 gives<br/>\\[\\log_{\\var{a}}(\\simplify{(x+{b})^2})- \\log_{\\var{a}}(\\simplify{(x+{c})})=\\log_{\\var{a}}\\left(\\simplify{(x+{b})^2/(x+{c})}\\right)\\]<br/>So the equation to solve becomes:<br/>\\[\\log_{\\var{a}}\\left(\\simplify{(x+{b})^2/(x+{c})}\\right)=\\var{d}\\]<br/>and using rule 3 this gives:<br/>\\[ \\begin{eqnarray} \\simplify{(x+{b})^2/(x+{c})}&amp;=&amp;{\\var{a}}^{\\var{d}}\\Rightarrow\\\\ \\simplify{(x+{b})^2}&amp;=&amp;{\\var{a}}^{\\var{d}}(\\simplify{x+{c}})=\\simplify{{a^d}(x+{c})}\\Rightarrow\\\\ \\simplify{x^2+{2*b-a^(d)}x+{b^2-a^(d)*c}}&amp;=&amp;0 \\end{eqnarray} \\]<br/>Solving this quadratic we get two solutions:</p>\n  <p>$x=\\var{sol1}$ and $x=\\var{sol2}$</p>\n  <p>We should check that these solutions gives positive values for $\\simplify{x+{b}}$ and $\\simplify{x+{c}}$ as otherwise the logs are not defined.</p>\n  <p><strong>The value $x=\\var{sol1}$ gives:&nbsp;</strong></p>\n  <p>Substituting this value for $x$ into $\\log_{\\var{a}}(\\simplify{x+{b}})$ we get $\\log_{\\var{a}}(\\simplify{{2*a^d}})$ so OK.</p>\n  <p>Substituting this value for $x$ into $\\log_{\\var{a}}(\\simplify{x+{c}})$ we get $\\log_{\\var{a}}(\\simplify{{4*a^d}})$ so OK.</p>\n  <p>Hence $x=\\var{sol1}$ is a solution to our original equation.</p>\n  <p><strong>The value $x=\\var{sol2}$ gives:</strong></p>\n  <p>Substituting this value for $x$ into $\\log_{\\var{a}}(\\simplify{x+{b}})$ we get $\\log_{\\var{a}}(\\simplify{{-a^d}})$ so <span class=\"caps\">NOT</span> OK.</p>\n  <p>Substituting this value for $x$ into $\\log_{\\var{a}}(\\simplify{x+{c}})$ we get $\\log_{\\var{a}}(\\simplify{{a^d}})$ so OK.</p>\n  <p>Hence $x=\\var{sol2}$ is <span class=\"caps\">NOT</span> a solution to our original equation as $\\log_{\\var{a}}(\\simplify{x+{b}})$ is not defined for this value of $x$.</p>\n  <p>So there is only one solution $x=\\var{sol1}$.</p>\n  ", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"]}, "parts": [{"stepspenalty": 1.0, "prompt": "\n      <p>\\[2\\log_{\\var{a}}(\\simplify{x+{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 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": "{sol1}", "type": "jme"}], "steps": [{"prompt": "\n          <p>Three rules for logs should be used:</p>\n          <p>1. $n\\log_a(m)=\\log_a(m^n)$</p>\n          <p>2. $\\log_a(b)-\\log_a(c)=\\log_a(b/c)$</p>\n          <p>3. $\\log_a(p)=r \\Rightarrow p=a^r$</p>\n          <p>So use rule 1 followed by rules 2 and 3 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(2,3)", "name": "a"}, "c": {"definition": "b+2*a^(d)", "name": "c"}, "b": {"definition": "s*random(1..20)", "name": "b"}, "d": {"definition": "random(1,2)", "name": "d"}, "s": {"definition": "random(1,-1)", "name": "s"}, "sol2": {"definition": "-c+a^d", "name": "sol2"}, "sol1": {"definition": "c-2*b", "name": "sol1"}}, "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": "\n            \t\t<p>Solve for $x$: $\\displaystyle 2\\log_{a}(x+b)- \\log_{a}(x+c)=d$.&nbsp;</p>\n            \t\t        <p>Make sure that your choice is a solution by substituting back into the equation.</p>\n            \t\t", "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/"}, {"name": "Wan Mekwi", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4058/"}]}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Wan Mekwi", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4058/"}]}