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{"name": "Logarithms: Simplifying Expressions 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Logarithms: Simplifying Expressions 2", "tags": [], "metadata": {"description": "<p>Rewriting expressions of the form $n\\log(a)\\pm m\\log(b)$ as a single logarithm.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Rewrite the following expression as a single logarithm:</p>\n<p>\\[\\simplify{{n1}log({a})+{n2}log({b})}.\\]</p>", "advice": "<p>{advice}</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"n1": {"name": "n1", "group": "Ungrouped variables", "definition": "random(2..3)", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(2..5 except a)", "description": "", "templateType": "anything", "can_override": false}, "advice": {"name": "advice", "group": "Ungrouped variables", "definition": "if(n2>0,'{advice1}','{advice2}')", "description": "", "templateType": "anything", "can_override": false}, "advice1": {"name": "advice1", "group": "Ungrouped variables", "definition": "\"<p>To rewrite $\\\\simplify{{n1}log({a})+{n2}log({b})}$ as a single logarithm, we want to use the following rules:</p>\\n<ul>\\n<li>$n\\\\log(a)=\\\\log(a^n)$;</li>\\n<li>$ \\\\log(a)+\\\\log(b) = \\\\log(ab) $.</li>\\n</ul>\\n<p>Hence,</p>\\n<p>\\\\[ \\\\begin{split} \\\\simplify{{n1}log({a})+{n2}log({b})} &amp;\\\\,= \\\\simplify[!collectNumbers]{log({a}^{n1})+log({b}^{n2})} \\\\\\\\ &amp;\\\\,= \\\\simplify{log({a^n1})+log({b^n2})} \\\\\\\\&amp;\\\\,=\\\\simplify{log({sol})}. \\\\end{split} \\\\]</p>\"", "description": "", "templateType": "long string", "can_override": false}, "advice2": {"name": "advice2", "group": "Ungrouped variables", "definition": "\"<p>To rewrite $\\\\simplify{{n1}log({a})+{n2}log({b})}$ as a single logarithm, we want to use the following rules:</p>\\n<ul>\\n<li>$n\\\\log(a)=\\\\log(a^n)$;</li>\\n<li>$ \\\\log(a)-\\\\log(b) = \\\\log\\\\left(\\\\frac{a}{b}\\\\right) $.</li>\\n</ul>\\n<p>Hence,</p>\\n<p>\\\\[ \\\\begin{split} \\\\simplify{{n1}log({a})+{n2}log({b})} &amp;\\\\,= \\\\simplify[!collectNumbers]{log({a}^{n1})-log({b}^{abs(n2)})} \\\\\\\\ &amp;\\\\,=\\\\simplify{log({a^n1})-log({b^abs(n2)})} \\\\\\\\ &amp;\\\\,=\\\\simplify[fractionNumbers]{log({sol})}. \\\\end{split} \\\\]</p>\"", "description": "", "templateType": "long string", "can_override": false}, "sol": {"name": "sol", "group": "Ungrouped variables", "definition": 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