// Numbas version: exam_results_page_options
{"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": {}, "ungrouped_variables": ["power", "d", "f", "g", "b", "a", "c", "ans"], "name": "Solving log equations using exponentials", "tags": ["exp", "exponential", "exponentials", "logarithm", "logarithms", "logs", "solving", "solving equations"], "advice": "", "rulesets": {}, "parts": [{"stepsPenalty": 0, "prompt": "<p>Solve the following equation for $x$</p>\n<p style=\"text-align: left;\">$\\displaystyle{\\var{d}\\log_\\var{b}(\\simplify{{a}x+{c}})+\\var{f}=\\var{g} &nbsp; }.$</p>\n<p style=\"text-align: center;\"></p>\n<p style=\"text-align: left;\">$x=$&nbsp;[[0]]</p>\n<p style=\"text-align: left;\"><strong>Note:&nbsp;</strong>You can use the symbol ^ to signify powers, and / to signify division. Please ensure you use brackets correctly.</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "fractionnumbers", "scripts": {}, "answer": "{({b}^{power}-{c})/{a}}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "vsetrange": [0, 1]}], "steps": [{"prompt": "<p>We start solving the equation one operation at a time by doing the inverse to both sides, when we get to undoing the log&nbsp;we recall the definition of $\\log_b$, write the equation in index form and continue solving.</p>\n<p><strong>Recall:</strong> The definition of $\\log_b$ says&nbsp;$\\log_b(a)=c$ is equivalent to $b^c=a$.</p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: right;\">$\\displaystyle{\\var{d}\\log_\\var{b}(\\simplify{{a}x+{c}})+\\var{f} &nbsp;}$</td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">$\\var{g}$</td>\n<td style=\"text-align: left;\"></td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">$\\displaystyle{\\var{d}\\log_\\var{b}(\\simplify{{a}x+{c}}) &nbsp;}$</td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">$\\var{g-f}$</td>\n<td style=\"text-align: left;\">(subtract $\\var{f}$ from both sides)&nbsp;</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">$\\displaystyle{\\log_\\var{b}(\\simplify{{a}x+{c}}) &nbsp;}$</td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">$\\var{power}$</td>\n<td style=\"text-align: left;\">(divide both sides by $\\var{d}$)</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">$\\simplify[basic]{{b}^{power}}$</td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">$\\simplify{{a}x+{c}}$</td>\n<td style=\"text-align: left;\">(using the definition of $\\log_\\var{b}$)</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">$\\simplify[basic,unitpower]{{b}^{power}-{c}}$</td>\n<td style=\"text-align: center;\">$=$&nbsp;</td>\n<td style=\"text-align: left;\">$\\var{a}x$</td>\n<td style=\"text-align: left;\">(subtract $\\var{c}$ from both sides)</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\"></td>\n<td style=\"text-align: center;\">&nbsp;</td>\n<td style=\"text-align: left;\"></td>\n<td style=\"text-align: left;\"></td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">$\\displaystyle{\\simplify[basic,fractionnumbers,unitpower]{({b}^{power}-{c})/{a}}}$</td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">$x$</td>\n<td style=\"text-align: left;\">(divide both sides by $\\var{a}$)</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\"></td>\n<td style=\"text-align: center;\">&nbsp;</td>\n<td style=\"text-align: left;\"></td>\n<td style=\"text-align: left;\"></td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">$x$</td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">$\\displaystyle{\\simplify[basic,fractionnumbers,unitpower]{{({b}^{power}-{c})/{a}}}}$</td>\n<td style=\"text-align: left;\"></td>\n</tr>\n</tbody>\n</table>\n<p></p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "scripts": {}, "marks": 0, "showCorrectAnswer": true, "type": "gapfill"}], "statement": "", "variable_groups": [{"variables": [], "name": "Unnamed group"}], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"a": {"definition": "random(2..12)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "random(-12..12 except 0)", "templateType": "anything", "group": "Ungrouped 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"question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "steve kilgallon", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/268/"}]}]}], "contributors": [{"name": "steve kilgallon", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/268/"}]}