// Numbas version: finer_feedback_settings
{"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": [{"name": "Solving exponential equations using logs", "tags": ["exp", "exponential", "exponentials", "logarithm", "Logarithm", "Logarithms", "logarithms", "Logs", "logs", "solving", "solving equations", "Solving equations"], "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "", "advice": "<p>a)&nbsp;</p>\n<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 $\\var{p}$)</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>\n<p>b)</p>\n<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;\">${\\var{FV}}$</td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">$\\displaystyle{\\simplify{{pay}((1+{int})^n-1)/{int}}}$</td>\n<td style=\"text-align: left;\"></td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">$\\simplify{{FV*int}}$</td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">$\\displaystyle{\\simplify{{pay}((1+{int})^n-1)}}$</td>\n<td style=\"text-align: left;\">(multiply both sides by&nbsp;$\\var{int}$)&nbsp;</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">$\\displaystyle{\\simplify[simplifyFractions]{{FV*int}/{pay}}}$</td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">$\\displaystyle{\\simplify{(1+{int})^n-1}}$</td>\n<td style=\"text-align: left;\">(divide both sides by $\\var{pay}$)</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">$\\displaystyle{\\simplify[simplifyFractions]{{FV*int}/{pay}+1}}$</td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">$\\displaystyle{\\simplify{(1+{int})^n}}$</td>\n<td style=\"text-align: left;\">(add $1$ to both sides)</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">$\\displaystyle{\\simplify[simplifyFractions]{{FV*int+pay}/{pay}}}$</td>\n<td style=\"text-align: center;\">$=$&nbsp;</td>\n<td style=\"text-align: left;\">$\\displaystyle{\\simplify{(1+{int})^n}}$</td>\n<td style=\"text-align: left;\">(tidy up left hand side)</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;\">$\\displaystyle{\\log\\left(\\simplify[simplifyFractions]{{FV*int+pay}/{pay}}\\right)}$</td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">$\\displaystyle{\\log\\left((\\var{1+int})^n\\right)}$</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;\"></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;\">$\\displaystyle{\\log\\left(\\simplify[simplifyFractions]{{FV*int+pay}/{pay}}\\right)}$</td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">$\\displaystyle{n\\log(\\var{1+int})}$</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;\">$\\displaystyle{\\frac{\\log\\left(\\simplify[simplifyFractions]{{FV*int+pay}/{pay}}\\right)}{\\log(\\var{1+int})}}$</td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">$n$</td>\n<td style=\"text-align: left;\">(divide both sides by $\\log(\\var{1+int})$)</td>\n</tr>\n</tbody>\n</table>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"test": {"name": "test", "group": "b", "definition": "fv*int/pay", "description": "", "templateType": "anything", "can_override": false}, "periods": {"name": "periods", "group": "b", "definition": "log(FV*int/pay+1)/log(1+int)", "description": "", "templateType": "anything", "can_override": false}, "logb": {"name": "logb", "group": "Ungrouped variables", "definition": "log(b)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(1.05..1.5#0.05)", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(1000..2000#20)", "description": "", "templateType": "anything", "can_override": false}, "logfrac": {"name": "logfrac", "group": "Ungrouped variables", "definition": "log(frac)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(20..a-10#20)", "description": "", "templateType": "anything", "can_override": false}, "frac": {"name": "frac", "group": "Ungrouped variables", "definition": "(a-c)/p", "description": "", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "d*logfrac/logb", "description": "", "templateType": "anything", "can_override": false}, "int": {"name": "int", "group": "b", "definition": "random(0.01..0.10#0.01)", "description": "", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(2,4,3,12,26,52)", "description": "", "templateType": "anything", "can_override": false}, "p": {"name": "p", "group": "Ungrouped variables", "definition": "random(2,5,10,20,(a-c)/2,(a-c)/5,(a-c)/10,(a-c)/20)", "description": "", "templateType": "anything", "can_override": false}, "pay": {"name": "pay", "group": "b", "definition": "random(100..2000#100)", "description": "", "templateType": "anything", "can_override": false}, "amc": {"name": "amc", "group": "Ungrouped variables", "definition": "a-c", "description": "", "templateType": "anything", "can_override": false}, "FV": {"name": "FV", "group": "b", "definition": "random(20000..100000#500)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "p", "c", "b", "frac", "logfrac", "logb", "d", "amc", "n"], "variable_groups": [{"name": "b", "variables": ["FV", "pay", "int", "periods", "test"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Solve the following equation for $n$</p>\n<p style=\"text-align: left;\">$\\begin{align*}\\simplify{{a}={p}({b})^(n/{d})+{c}}.\\end{align*}$</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>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{d}*log({frac})/log({b})", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Solve the following equation for $n$</p>\n<p style=\"text-align: left;\">$\\displaystyle{\\simplify{{FV}={pay}((1+{int})^n-1)/{int}}}.$</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>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "log({FV*int+pay}/{pay})/log({1+int})", "answerSimplification": "simplifyFractions", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question", "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}]}]}], "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}]}