// Numbas version: exam_results_page_options
{"name": "David's copy of 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": [{"parts": [{"steps": [{"type": "information", "marks": 0, "variableReplacements": [], "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "scripts": {}, "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>", "showFeedbackIcon": true}], "gaps": [{"type": "jme", "vsetrangepoints": 5, "variableReplacements": [], "checkvariablenames": false, "showCorrectAnswer": true, "answersimplification": "fractionnumbers", "variableReplacementStrategy": "originalfirst", "scripts": {}, "showpreview": true, "checkingaccuracy": 0.001, "vsetrange": [0, 1], "marks": 1, "answer": "{({b}^{power}-{c})/{a}}", "expectedvariablenames": [], "checkingtype": "absdiff", "showFeedbackIcon": true}], "marks": 0, "variableReplacements": [], "type": "gapfill", "scripts": {}, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "stepsPenalty": "1", "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>", "showFeedbackIcon": true}], "functions": {}, "preamble": {"css": "", "js": ""}, "ungrouped_variables": ["power", "d", "f", "g", "b", "a", "c", "ans"], "name": "David's copy of Solving log equations using exponentials", "advice": "", "variables": {"c": {"definition": "random(-12..12 except 0)", "templateType": "anything", "name": "c", "group": "Ungrouped variables", "description": ""}, "g": {"definition": "f+power*d", "templateType": "anything", "name": "g", "group": "Ungrouped variables", "description": ""}, "a": {"definition": "random(2..12)", "templateType": "anything", "name": "a", "group": "Ungrouped variables", "description": ""}, "power": {"definition": "random(-5..5 except 0)", "templateType": "anything", "name": "power", "group": "Ungrouped variables", "description": ""}, "d": {"definition": "random(2..12)", "templateType": "anything", "name": "d", "group": "Ungrouped variables", "description": ""}, "f": {"definition": "random(1..60)", "templateType": "anything", "name": "f", "group": "Ungrouped variables", "description": ""}, "ans": {"definition": "(b^((g-f)/d)-c)/a", "templateType": "anything", "name": "ans", "group": "Ungrouped variables", "description": ""}, "b": {"definition": "random(2,3,4,5,10)", "templateType": "anything", "name": "b", "group": "Ungrouped variables", "description": ""}}, "tags": ["exp", "exponential", "exponentials", "logarithm", "logarithms", "logs", "solving", "Solving equations", "solving equations"], "statement": "", "variable_groups": [{"variables": [], "name": "Unnamed group"}], "rulesets": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "metadata": {"licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International", "description": ""}, "extensions": [], "type": "question", "contributors": [{"name": "David Martin", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/130/"}, {"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}]}]}], "contributors": [{"name": "David Martin", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/130/"}, {"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}]}