// Numbas version: exam_results_page_options
{"name": "Logarithms: Solving equations 6", "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 6", "tags": ["logarithmic equations", "logarithms", "logs", "quadratic equation", "solving a quadratic equation", "solving equations involving powers", "solving logarithmic equations"], "advice": "\n    <p>In the equation<br/>\\[\\var{c}\\left(\\var{a^2}^x \\right)+ \\var{d}\\left(\\var{a}^{x+1}\\right)=\\var{b}\\]</p>\n  <p>Let $y=\\var{a}^x$ and since $\\var{a^2}^x=(\\var{a}^2)^x=\\left(\\var{a}^x\\right)^2 $ on substitution this becomes:<br/>\\[\\begin{eqnarray*} \\var{c}y^2 + \\var{d}\\left(\\var{a}y\\right)&amp;=&amp;\\var{b} \\\\&nbsp; \\\\ \\Rightarrow \\var{c}y^2 + \\var{a*d}y&amp;=&amp;\\var{b}\\Rightarrow \\var{c}y^2 + \\var{a*d}y-\\var{b}=0\\\\ \\\\ \\Rightarrow (\\var{c}y+\\var{al})(y-\\var{abs(be)})&amp;=&amp;0 \\mbox{ on factorisation.} \\end{eqnarray*} \\]<br/>This quadratic has solutions $\\displaystyle y=\\simplify[std]{{-al}/{c}},\\;\\;y=\\var{-be}$.</p>\n  <p>But $y=\\var{a}^x \\gt 0$ for all $x$ and so $\\displaystyle y=\\simplify[std]{{-al}/{c}}$ cannot be a solution for the original equation.</p>\n  <p>We are left with $y=\\var{-be}$ which gives:<br/>\\[\\begin{eqnarray*} \\var{a}^x &amp;=&amp; \\var{-be}\\\\ \\Rightarrow x\\ln(\\var{a})&amp;=&amp;\\ln(\\var{-be})\\\\ \\Rightarrow x&amp;=&amp;\\frac{\\ln(\\var{-be})}{\\ln(\\var{a})} = \\var{ans1} \\end{eqnarray*} \\]</p>\n  <p>to 3 decimal places.</p>\n    ", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"prompt": "\n        <p>$x=\\;\\;$[[0]]</p>\n      <p>Input your answer to 3 decimal places.</p>\n        ", "gaps": [{"minvalue": "ans1-tol", "type": "numberentry", "maxvalue": "ans1+tol", "marks": 2.0, "showPrecisionHint": false}], "type": "gapfill", "marks": 0.0}], "extensions": [], "statement": "\n    <p>Solve the following equation for $x$. Note that there is only one solution.</p>\n    <p>\\[\\var{c}\\left(\\var{a^2}^x \\right)+ \\var{d}\\left(\\var{a}^{x+1}\\right)=\\var{b}\\]</p>\n    <p>Hint: remember that $\\left(A^2\\right)^x=\\left(A^x\\right)^2$ for any number $A$.</p>\n    ", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "random(4..9)", "name": "a"}, "be": {"definition": "random(-9..-2)", "name": "be"}, "c": {"definition": "random(2..6)", "name": "c"}, "b": {"definition": "-al*be", "name": "b"}, "d": {"definition": "if(c=2 or c=4,random(3,5,7,9),if(c=3,random(2,4,5,7,8),if(c=5,random(2,3,4,6,7),random(5,7,11))))", "name": "d"}, "ans1": {"definition": "precround(tans1,3)", "name": "ans1"}, "al": {"definition": "d*a-be*c", "name": "al"}, "tol": {"definition": 0.0, "name": "tol"}, "tans1": {"definition": "ln(-be)/ln(a)", "name": "tans1"}}, "metadata": {"notes": "<p><strong>2/07/2012:</strong></p>\n            <p>Added tags.</p>\n            <p>Added that the solution is to 3 decimal places.</p>\n            <p>Forced exact solution to 3 decimal places - no tolerance.</p>\n            <p>Improved display.</p>\n            <p><strong>19/07/2012:</strong></p>\n            <p>Added description.</p>\n            <p>Checked calculation.</p>\n            <p>New tolerance variable tol=0 for the answer.</p>\n            <p>&nbsp;</p>\n            <p><strong>25/07/2012:</strong></p>\n            <p>Added tags.</p>\n            <p>Is it necessary to include the hint? It is a rather basic mathematical identity.</p>\n            <p>&nbsp;</p>\n            <p>In the Advice section moved \\Rightarrow so that it is at the beginning of the line instead of the end of the previous line.</p>\n            <p>&nbsp;</p>\n            <p>&nbsp;</p>\n            <p>Question appears to be working correctly.</p>\n            <p>&nbsp;</p>", "description": "<p>Solve for $x$: $c(a^2)^x + d(a)^{x+1}=b$ (there is only one solution for this example).</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/"}, {"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/"}]}