// Numbas version: exam_results_page_options
{"name": "Solving exponential equations using logs", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["a", "p", "c", "b", "frac", "logfrac", "logb", "d", "amc", "n"], "name": "Solving exponential equations using logs", "tags": ["exp", "exponential", "exponentials", "logarithm", "logarithms", "logs", "solving", "solving equations"], "preamble": {"css": "", "js": ""}, "advice": "", "rulesets": {}, "parts": [{"stepsPenalty": 0, "prompt": "<p>Solve the following equation for $n$</p>\n<p style=\"text-align: left;\">$\\displaystyle{\\simplify{{a}={p}({b})^(n/{d})+{c}}}.$</p>\n<p style=\"text-align: center;\"></p>\n<p style=\"text-align: left;\">$n=$&nbsp;[[0]]</p>\n<p style=\"text-align: left;\"><strong></strong></p>\n<p style=\"text-align: left;\"><strong>Note:</strong><span>&nbsp;Typing $\\log(5)$&nbsp;will input the value&nbsp;$\\log_{10}(5)$, whereas&nbsp;$\\log5$ will not work.<br/><strong>Note:&nbsp;</strong>Typing $\\ln(5)$ will input the value $\\log_e(5)$, whereas $\\ln5$ will not work.</span></p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "{d}*log({frac})/log({b})", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "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 exponential we apply a log to both sides, we then use a log law and continue solving.</p>\n<p></p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: right;\">$\\var{a}$</td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">$\\simplify{{p}({b})^(n/{d})+{c}}$</td>\n<td style=\"text-align: left;\"></td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">$\\simplify{{a-c}}$</td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">$\\simplify{{p}({b})^(n/{d})}$</td>\n<td style=\"text-align: left;\">(subtract $\\var{c}$ from both sides)&nbsp;</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">$\\var{frac}$</td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">$\\simplify{{b}^(n/{d})}$</td>\n<td style=\"text-align: left;\">(divide both sides by $\\var{a}$)</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">$\\log(\\var{frac})$</td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">$\\log(\\var{b}^{\\frac{n}{\\var{d}}})$</td>\n<td style=\"text-align: left;\">(take the log of 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;\">$\\frac{n}{\\var{d}}\\log(\\var{b})$</td>\n<td style=\"text-align: left;\">(use a log law)</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\"></td>\n<td style=\"text-align: center;\"></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;\">$\\displaystyle{\\frac{\\log(\\var{frac})}{\\log(\\var{b})}}$</td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">$\\frac{n}{\\var{d}}$</td>\n<td style=\"text-align: left;\">(divide both sides by $\\log(\\var{b})$)</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\"></td>\n<td style=\"text-align: center;\"></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;\">$\\displaystyle{\\frac{\\var{d}\\log(\\var{frac})}{\\log(\\var{b})}}$</td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">$n$</td>\n<td style=\"text-align: left;\">(multiply both sides by $\\var{d}$)</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": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "random(1000..2000#20)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "random(20..a-10#20)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "random(1.05..1.5#0.05)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "frac": {"definition": "(a-c)/p", "templateType": "anything", "group": "Ungrouped variables", "name": "frac", "description": ""}, "logfrac": {"definition": "log(frac)", "templateType": "anything", "group": "Ungrouped variables", "name": "logfrac", "description": ""}, "amc": {"definition": "a-c", "templateType": "anything", "group": "Ungrouped variables", "name": "amc", "description": ""}, "n": {"definition": "d*logfrac/logb", "templateType": "anything", "group": "Ungrouped variables", "name": "n", "description": ""}, "p": {"definition": "random(2,5,10,20,(a-c)/2,(a-c)/5,(a-c)/10,(a-c)/20)", "templateType": "anything", "group": "Ungrouped variables", "name": "p", "description": ""}, "logb": {"definition": "log(b)", "templateType": "anything", "group": "Ungrouped variables", "name": "logb", "description": ""}, "d": {"definition": "random(2,4,3,12,26,52)", "templateType": "anything", "group": "Ungrouped variables", "name": "d", "description": ""}}, "metadata": {"notes": "", "description": "", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "type": "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/"}]}