// 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": [{"name": "Solving log equations using exponentials", "advice": "", "variable_groups": [{"variables": [], "name": "Unnamed group"}], "ungrouped_variables": ["power", "d", "f", "g", "b", "a", "c", "ans"], "functions": {}, "extensions": [], "variablesTest": {"condition": "", "maxRuns": 100}, "type": "question", "statement": "", "preamble": {"js": "", "css": ""}, "tags": ["exp", "exponential", "exponentials", "logarithm", "logarithms", "logs", "solving", "Solving equations", "solving equations"], "parts": [{"showFeedbackIcon": true, "marks": 0, "scripts": {}, "type": "gapfill", "showCorrectAnswer": true, "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>", "steps": [{"showFeedbackIcon": true, "marks": 0, "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>", "type": "information", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true}], "gaps": [{"scripts": {}, "checkingtype": 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