// Numbas version: finer_feedback_settings
{"name": "Solving log equations using exponentials", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "advice": "", "ungrouped_variables": ["power", "d", "f", "g", "b", "a", "c", "ans"], "variable_groups": [{"variables": [], "name": "Unnamed group"}], "preamble": {"css": "", "js": ""}, "tags": ["exp", "exponential", "exponentials", "logarithm", "logarithms", "logs", "solving", "Solving equations", "solving equations"], "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "rulesets": {}, "variables": {"d": {"description": "", "definition": "random(2..12)", "templateType": "anything", "name": "d", "group": "Ungrouped variables"}, "power": {"description": "", "definition": "random(-5..5 except 0)", "templateType": "anything", "name": "power", "group": "Ungrouped variables"}, "ans": {"description": "", "definition": "(b^((g-f)/d)-c)/a", "templateType": "anything", "name": "ans", "group": "Ungrouped variables"}, "a": {"description": "", "definition": "random(2..12)", "templateType": "anything", "name": "a", "group": "Ungrouped variables"}, "g": {"description": "", "definition": "f+power*d", "templateType": "anything", "name": "g", "group": "Ungrouped variables"}, "b": {"description": "", "definition": "random(2,3,4,5,10)", "templateType": "anything", "name": "b", "group": "Ungrouped variables"}, "f": {"description": "", "definition": "random(1..60)", "templateType": "anything", "name": "f", "group": "Ungrouped variables"}, "c": {"description": "", "definition": "random(-12..12 except 0)", "templateType": "anything", "name": "c", "group": "Ungrouped variables"}}, "name": "Solving log equations using exponentials", "statement": "", "parts": [{"gaps": [{"checkvariablenames": false, "answer": "{({b}^{power}-{c})/{a}}", "marks": 1, "checkingtype": "absdiff", "showpreview": true, "scripts": {}, "checkingaccuracy": 0.001, "variableReplacements": [], "vsetrangepoints": 5, "type": "jme", "answersimplification": "fractionnumbers", "showFeedbackIcon": true, "showCorrectAnswer": true, "expectedvariablenames": [], "vsetrange": [0, 1], "variableReplacementStrategy": "originalfirst"}], "type": "gapfill", "showFeedbackIcon": true, "showCorrectAnswer": true, "marks": 0, "stepsPenalty": "1", "steps": [{"type": "information", "showFeedbackIcon": true, "showCorrectAnswer": true, "marks": 0, "prompt": "We start solving the equation one operation at a time by doing the inverse to both sides, when we get to undoing the log we recall the definition of $\\log_b$, write the equation in index form and continue solving.
\nRecall: The definition of $\\log_b$ says $\\log_b(a)=c$ is equivalent to $b^c=a$.
\n\n\n\n| $\\displaystyle{\\var{d}\\log_\\var{b}(\\simplify{{a}x+{c}})+\\var{f} }$ | \n$=$ | \n$\\var{g}$ | \n | \n
\n\n| $\\displaystyle{\\var{d}\\log_\\var{b}(\\simplify{{a}x+{c}}) }$ | \n$=$ | \n$\\var{g-f}$ | \n(subtract $\\var{f}$ from both sides) | \n
\n\n| $\\displaystyle{\\log_\\var{b}(\\simplify{{a}x+{c}}) }$ | \n$=$ | \n$\\var{power}$ | \n(divide both sides by $\\var{d}$) | \n
\n\n| $\\simplify[basic]{{b}^{power}}$ | \n$=$ | \n$\\simplify{{a}x+{c}}$ | \n(using the definition of $\\log_\\var{b}$) | \n
\n\n| $\\simplify[basic,unitpower]{{b}^{power}-{c}}$ | \n$=$ | \n$\\var{a}x$ | \n(subtract $\\var{c}$ from both sides) | \n
\n\n | \n | \n | \n | \n
\n\n| $\\displaystyle{\\simplify[basic,fractionnumbers,unitpower]{({b}^{power}-{c})/{a}}}$ | \n$=$ | \n$x$ | \n(divide both sides by $\\var{a}$) | \n
\n\n | \n | \n | \n | \n
\n\n| $x$ | \n$=$ | \n$\\displaystyle{\\simplify[basic,fractionnumbers,unitpower]{{({b}^{power}-{c})/{a}}}}$ | \n | \n
\n\n
\n", "variableReplacementStrategy": "originalfirst", "scripts": {}, "variableReplacements": []}], "prompt": "Solve the following equation for $x$
\n$\\displaystyle{\\var{d}\\log_\\var{b}(\\simplify{{a}x+{c}})+\\var{f}=\\var{g} }.$
\n\n$x=$ [[0]]
\nNote: You can use the symbol ^ to signify powers, and / to signify division. Please ensure you use brackets correctly.
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