// Numbas version: exam_results_page_options
{"name": "Logarithms: Solving equations 4", "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 4", "tags": ["equations solved by using logarithms", "laws of logarithms", "logarithm laws", "logarithm rules", "logarithmic expressions", "logarithms", "solving equations by taking logarithms", "solving logarithmic equations"], "advice": "<p>Both parts of this question can be solved in a similar way, by taking logarithms of both sides of each equation.</p>\n  <p>a)</p>\n  <p>Taking logs (to the base 10 in this case &ndash; but any base will do) of both sides of</p>\n  <p>\\[\\var{n}^{\\simplify{{a1}*x+{b1}}}=\\var{m}^{\\var{c1}x}\\]</p>\n  <p>gives on using the rule $\\log(a^b)=b\\log(a)$:</p>\n  <p>\\[\\begin{eqnarray*} \\simplify[std]{({a1}x+{b1})log({n})}&amp;=&amp;\\simplify[std]{{c1}*x*log({m})}\\\\ \\Rightarrow\\simplify[std]{x({a1}*log({n})-{c1}*log({m}))} &amp;=&amp;\\var{-b1}\\log(\\var{n})\\\\ \\Rightarrow x&amp;=&amp;\\simplify[std]{({-b1}*log({n}))/({a1}log({n})-{c1}*log({m}))}\\\\ &amp;=&amp; \\var{ans1}\\mbox{ to 3 decimal places} \\end{eqnarray*} \\]</p>\n  <p>b)</p>\n  <p>Similarly, taking logs of both sides of:<br/>\\[\\var{a2}^{\\var{b2}x^2}=\\var{c2}^{\\var{d2}x}\\]</p>\n  <p>gives:</p>\n  <p>\\[ \\simplify[std]{({b2}x^2)log({a2})}=\\simplify[std]{{d2}*x*log({c2})} \\Rightarrow \\simplify[std]{x({b2}*log({a2})x-{d2}*log({c2}))} =0\\]</p>\n  <p>and so the solutions are:</p>\n  <p>1. $x=0$</p>\n  <p>or</p>\n  <p>2. $\\displaystyle x=\\simplify[std,!fractionNumbers]{({d2}*log({c2}))/({b2}*log({a2})) = {ans2}}$ to 3 decimal places.</p>", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"prompt": "\n        <p>\\[\\var{n}^{\\simplify{{a1}*x+{b1}}}=\\var{m}^{\\var{c1}x}\\]</p>\n        <p>$x=\\;\\;$[[0]].</p>\n        <p>Enter your answer to 3 decimal places.</p>\n        ", "gaps": [{"minvalue": "{ans1}", "type": "numberentry", "maxvalue": "{ans1}", "marks": 1.0, "showPrecisionHint": false}], "type": "gapfill", "marks": 0.0}, {"prompt": "\n        <p>\\[\\var{a2}^{\\var{b2}x^2}=\\var{c2}^{\\var{d2}x}\\]</p>\n        <p>$x=\\;\\;$[[0]] $\\;\\;\\;$ (Enter the smallest value of $x$ here).</p>\n        <p>$x=\\;\\;$[[1]] $\\;\\;\\;$ (Enter the largest value of $x$ here).</p>\n        <p>Enter your answers to 3 decimal places.</p>\n        ", "gaps": [{"minvalue": 0.0, "type": "numberentry", "maxvalue": 0.0, "marks": 0.5, "showPrecisionHint": false}, {"minvalue": "ans2", "type": "numberentry", "maxvalue": "ans2", "marks": 1.5, "showPrecisionHint": false}], "type": "gapfill", "marks": 0.0}], "extensions": [], "statement": "<p>Find all values of $x$ that satisfy the following equations:</p>", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"tc2": {"definition": "random(2..9)", "name": "tc2"}, "s3": {"definition": "random(1,-1)", "name": "s3"}, "ans1": {"definition": "precround(tans1,3)", "name": "ans1"}, "s1": {"definition": "random(1,-1)", "name": "s1"}, "m": {"definition": "if(tm=n,17,tm)", "name": "m"}, "d2": {"definition": "random(2..9)", "name": "d2"}, "n": {"definition": "random(2,3,5,7,11,13)", "name": "n"}, "a1": {"definition": "s1*random(1..9)", "name": "a1"}, "tm": {"definition": "random(2,3,5,7,11,13)", "name": "tm"}, "b1": {"definition": "random(1..9)", "name": "b1"}, "b2": {"definition": "random(2..9)", "name": "b2"}, "c2": {"definition": "if(tc2=a2,tc2+1,tc2)", "name": "c2"}, "c1": {"definition": "s3*random(2..9)", "name": "c1"}, "tans1": {"definition": "-b1*log(n)/(a1*log(n)-c1*log(m))", "name": "tans1"}, "ans2": {"definition": "precround(tans2,3)", "name": "ans2"}, "a2": {"definition": "random(2..9)", "name": "a2"}, "tans2": {"definition": "d2*log(c2)/(b2*log(a2))", "name": "tans2"}}, "metadata": {"notes": "\n                        \t\t<p><strong>2/06/2012:</strong></p>\n                        \t\t<p>Added tags.</p>\n                        \t\t<p>Improved display.</p>\n                        \t\t<p>Forced solution in second part to be accurate to 3 decimal places with no tolerance.</p>\n                        \t\t<p>&nbsp;<strong>19/07/2012:</strong></p>\n                        \t\t<p>Added description.</p>\n                        \t\t<p>Checked calculation.</p>\n                        \t\t<p><strong>25/07/2012:</strong></p>\n                        \t\t<p>Added tags.</p>\n                        \t\t<p>Corrected a typo.</p>\n                        \t\t<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                        \t\t<p>Question appears to be working correctly.</p>\n                        \t\t<p>&nbsp;</p>\n                        \t\t<p>&nbsp;</p>\n                        \t\t", "description": "<p>Solve for $x$ each of the following equations:&nbsp;$n^{ax+b}=m^{cx}$ and $p^{rx^2}=q^{sx}$.</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/"}]}