// Numbas version: exam_results_page_options
{"name": "Logarithms: Solving equations 3", "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 3", "tags": ["exponential", "exponentiation", "laws of logarithms", "laws of logs", "log laws", "logarithm laws", "logarithm rules", "logarithms", "logs", "solving equations", "solving logarithmic equations"], "advice": "\n    <p>First use one of the logarith laws which states (for logarithms to any base)</p>\n    <p>\\[\\log(a)-\\log(b)=\\log\\left(\\frac{a}{b}\\right)\\]</p>\n    <p>So the equation can be written as:</p>\n    <p>\\[\\log_{10}\\left(\\simplify[std]{({a1}x+{b1})/({c1}x+{d1})}\\right)=\\var{e1}\\]<br/>Now exponentiate both sides to get:<br/>\\[\\simplify[std,!otherNumbers]{({a1}x+{b1})/({c1}x+{d1})}=10^{\\var{e1}} \\Rightarrow \\simplify[std,!otherNumbers]{{a1}x+{b1}=10^{e1}({c1}x+{d1})}\\]<br/>Collect together terms in $x$ on the left and everything else on the right of the equation gives:<br/>\\[\\simplify[std,!otherNumbers]{x({a1}-10^{e1}*{c1})=10^{e1}*{d1}-{b1}}\\]<br/>Finally rearrange to get:<br/>\\[\\simplify[std]{x=(10^{e1}*{d1}-{b1})/({a1}-10^{e1}*{c1})={10^e1*d1-b1}/{a1-10^e1*c1}}\\]<br/>which to 3 decimal places evaluates to <br/>\\[x=\\var{ans}.\\]</p>\n    ", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"prompt": "\n        <p>Input the solution for $x$ here:</p>\n      <p>$x=\\;\\;$ [[0]]</p>\n      <p>Input your answer to 3 decimal places.</p>\n        ", "gaps": [{"minvalue": "ans-tol", "type": "numberentry", "maxvalue": "ans+tol", "marks": 1.0, "showPrecisionHint": false}], "type": "gapfill", "marks": 0.0}], "extensions": [], "statement": "\n    <p>Solve the following equation for $x$.</p>\n  <p>\\[\\simplify[std]{log({a1}x+{b1})-log({c1}x+{d1})={e1}}\\]</p>\n    ", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"e1": {"definition": "random(1..2)", "name": "e1"}, "a1": {"definition": "random(1..9)", "name": "a1"}, "b1": {"definition": "9+random(1..9)", "name": "b1"}, "tol": {"definition": 0.0, "name": "tol"}, "ans": {"definition": "precround(tans,3)", "name": "ans"}, "c1": {"definition": "d1*random(1..9)", "name": "c1"}, "tans": {"definition": "(d1*10^e1-b1)/(a1-c1*10^e1)", "name": "tans"}, "d1": {"definition": "random(1..9)", "name": "d1"}}, "metadata": {"notes": "\n                \t\t<p><strong>2/07/2012:</strong></p>\n                \t\t<p>Added tags.</p>\n                \t\t<p>Solution to 3 decimal places - no tolerance via new tolerance variable tol=0.</p>\n                \t\t<p>Improved display of Advice.</p>\n                \t\t<p><strong>19/07/2012:</strong></p>\n                \t\t<p>Added description.</p>\n                \t\t<p>Checked calculation.</p>\n                \t\t<p>&nbsp;</p>\n                \t\t<p><strong>25/07/2012:</strong></p>\n                \t\t<p>Added tags.</p>\n                \t\t<p>Removed a stray full stop.</p>\n                \t\t<p>Question appears to be working correctly.</p>\n                \t\t<p>&nbsp;</p>\n                \t\t", "description": "<p>Solve for $x$: $\\log(ax+b)-\\log(cx+d)=s$</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/"}]}