// Numbas version: exam_results_page_options {"name": "Logarithms: Solving equations 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variables": {"b": {"templateType": "anything", "name": "b", "description": "", "definition": "random(1..20)", "group": "Ungrouped variables"}, "c": {"templateType": "anything", "name": "c", "description": "", "definition": "b-random(1..20)", "group": "Ungrouped variables"}, "a": {"templateType": "anything", "name": "a", "description": "", "definition": "random(2..5)", "group": "Ungrouped variables"}, "d": {"templateType": "anything", "name": "d", "description": "", "definition": "random(1,2)", "group": "Ungrouped variables"}}, "statement": "\n
Solve the following equation for $x$.
\nInput your answer as a fraction or an integer as appropriate and not as a decimal.
\n ", "name": "Logarithms: Solving equations 1", "parts": [{"marks": 0, "showCorrectAnswer": true, "scripts": {}, "steps": [{"marks": 0, "showCorrectAnswer": true, "scripts": {}, "type": "information", "prompt": "Two rules for logs should be used:
\n1. $\\log_a(\\frac{x}{y})=\\log_a(x)-\\log_a(y)$
\n2. $a^x=y \\iff \\log_a y=x$
\nUse rule 1 followed by rule 2 to get an equation for $x$.
", "variableReplacementStrategy": "originalfirst", "variableReplacements": []}], "prompt": "\n\\[\\log_{\\var{a}}(x+\\var{b})- \\log_{\\var{a}}(\\simplify{(x+{c})})=\\var{d}\\]
\n$x=\\;$ [[0]]
\nIf you want help in solving the equation, click on Show steps. If you do so then you will lose 1 mark.
\nInput all numbers as fractions or integers and not as decimals.
\n ", "gaps": [{"answersimplification": "std", "vsetrangepoints": 5, "variableReplacements": [], "marks": 2, "showpreview": true, "variableReplacementStrategy": "originalfirst", "checkingtype": "absdiff", "notallowed": {"showStrings": false, "message": "Input as a fraction or an integer, not as a decimal.
", "strings": ["."], "partialCredit": 0}, "answer": "{b-c*a^d}/{a^d-1}", "showCorrectAnswer": true, "scripts": {}, "type": "jme", "vsetrange": [0, 1], "checkvariablenames": false, "checkingaccuracy": 0.0001, "expectedvariablenames": []}], "variableReplacements": [], "stepsPenalty": 1, "variableReplacementStrategy": "originalfirst", "type": "gapfill"}], "functions": {}, "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "variable_groups": [], "preamble": {"css": "", "js": ""}, "tags": ["algebra", "algebraic manipulation", "combining logarithms", "logarithm laws", "logarithms", "rebel", "rebelmaths", "simplifying logarithms", "solving equations", "Solving equations", "steps", "Steps"], "question_groups": [{"name": "", "pickQuestions": 0, "questions": [], "pickingStrategy": "all-ordered"}], "variablesTest": {"maxRuns": 100, "condition": ""}, "ungrouped_variables": ["a", "c", "b", "d"], "advice": "We use the following two rules for logs :
\n1. $\\log_a(\\frac{x}{y})=\\log_a(x)-\\log_a(y)$
\n2. $a^x=y \\iff \\log_a y=x$
\nUsing rule 1 we get
\\[\\log_{\\var{a}}(x+\\var{b})- \\log_{\\var{a}}(\\simplify{(x+{c})})=\\log_{\\var{a}}\\left(\\simplify{(x+{b})/(x+{c})}\\right)\\]
So the equation to solve becomes:
\\[\\log_{\\var{a}}\\left(\\simplify{(x+{b})/(x+{c})}\\right)=\\var{d}\\]
and using rule 2 this gives:
\\[ \\begin{eqnarray} \\simplify{(x+{b})/(x+{c})}&=&{\\var{a}}^{\\var{d}}\\Rightarrow\\\\ x+\\var{b}&=&{\\var{a}}^{\\var{d}}(x+\\var{c})=\\simplify{{a^d}}(x+\\var{c})\\Rightarrow\\\\ \\simplify{{a^d-1}x}&=&\\simplify[std]{{b}-{c}*{a^d}={b-c*a^d}}\\Rightarrow\\\\ x&=&\\simplify{{b-c*a^d}/{a^d-1}} \\end{eqnarray} \\]
We should check that this solution gives positive values for $x+\\var{b}$ and $\\simplify{x+{c}}$ as otherwise the logs are not defined.
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.
\nFor $\\log_{\\var{a}}(\\simplify{x+{c}})$ we get on substituting for $x$, $\\log_{\\var{a}}(\\simplify{({b-c })/{a^d-1}})$ so OK.
\nHence the value we found for $x$ is a solution to the original equation.
", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "Solve for $x$: $\\log_{a}(x+b)- \\log_{a}(x+c)=d$
", "notes": "5/08/2012:
\nAdded tags.
\nAdded description.
\nChecked calculation.OK.
\nImproved display in content areas.
\nrebelmaths rebel Rebel REBEL
"}, "showQuestionGroupNames": false, "type": "question", "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}]}], "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}