The rules for combining logs are
\n \n \n \n\\[\\begin{eqnarray*}\n \n \\log_a(bc)&=&\\log_a(b)+\\log_a(c)\\\\\n \n \\\\\n \n \\log_a\\left(\\frac{b}{c}\\right)&=&\\log_a(b)-\\log_a(c)\\\\\n \n \\\\\n \n \\log_a(b^r)&=&r\\log_a(b)\n \n \\end{eqnarray*}\n \n \\]
\n \n \n \na)
Using these rules gives:
\\[ \\begin{eqnarray*}\n \n \\log_a(x^{\\var{a1}}y^{\\var{b1}})&=&\\log_a(x^{\\var{a1}})+\\log_a(y^{\\var{b1}})\\\\\n \n &=&\\var{a1}\\log_a(x)+\\var{b1}\\log_a(y)\n \n \\end{eqnarray*}\n \n \\]
b)
\\[\\begin{eqnarray*}\n \n \\simplify[std]{{a2}/{b2}}\\log_a(x)+\\log_a(\\simplify{{c}*x+{d}})-\\log_a(\\simplify{x^(1/{b2})})&=&\\log_a(x^\\frac{\\var{a2}}{\\var{b2}})+\\log_a(\\simplify{{c}*x+{d}})-\\log_a(\\simplify{x^(1/{b2})})\\\\\n \n \\\\\n \n &=&\\log_a\\left(\\simplify[std]{(x^({a2}/{b2})*({c}x+{d}))/(x^(1/{b2}))}\\right)\\\\\n \n &=&\\log_a\\left(\\simplify{x^{f}*({c}x+{d})}\\right)\n \n \\end{eqnarray*}\n \n \\]
Express the following in terms of $\\log_a(x)$ and $\\log_a(y)$
\n\\[\\log_a(x^{\\var{a1}}y^{\\var{b1}})=\\alpha\\log_a(x)+\\beta\\log_a(y)\\]
\n$\\alpha=\\;\\;$[[0]], $\\beta=\\;\\;$[[1]]
\nIn Show steps you will find a video explaining the rules of logarithms by going through simplification of logs of numbers rather than algebraic expressions.
", "gaps": [{"minvalue": "{a1}", "type": "numberentry", "maxvalue": "{a1}", "marks": 0.5, "showPrecisionHint": false}, {"minvalue": "{b1}", "type": "numberentry", "maxvalue": "{b1}", "marks": 0.5, "showPrecisionHint": false}], "steps": [{"prompt": "This video covers the standard rules for logarithms.
\nThese are, for any base $b \\gt 0$:
\n\\[\\begin{eqnarray*} &1.& \\log_b(p)+\\log_b(q)&=&\\log_b(p\\times q)\\\\&2.& \\log_b(p)-\\log_b(q)&=&\\log_b\\left(\\frac{p}{ q}\\right)\\\\ &3.& \\log_b(p^r)=r\\log_b(p)\\\\&4.& \\log_b(1)=0\\end{eqnarray*} \\]
\n\n
", "type": "information", "marks": 0.0}], "marks": 0.0, "type": "gapfill"}, {"prompt": "\n
\\[\\simplify[std]{{a2}/{b2}}\\log_a(x)+\\log_a(\\simplify{{c}*x+{d}})-\\log_a(\\simplify{x^(1/{b2})})=\\log_a(q(x))\\]
\n$q(x)=\\;\\;$[[0]]
\n \n ", "gaps": [{"checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 1.0, "answer": "((x ^ {f}) * (({c} * x) + {d}))", "type": "jme"}], "type": "gapfill", "marks": 0.0}], "extensions": [], "statement": "Answer the following questions on logarithms.
\n", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"c": {"definition": "random(1..9)", "name": "c"}, "d": {"definition": "s4*random(2..9)", "name": "d"}, "f": {"definition": "precround((a2-1)/b2,0)", "name": "f"}, "s1": {"definition": "random(1,-1)", "name": "s1"}, "s4": {"definition": "random(1,-1)", "name": "s4"}, "a1": {"definition": "s1*random(2..9)", "name": "a1"}, "a2": {"definition": "1+b2*random(2..5)", "name": "a2"}, "b1": {"definition": "random(2..15)", "name": "b1"}, "b2": {"definition": "random(2..9)", "name": "b2"}}, "metadata": {"notes": "\n \t\t
2/06/2012:
\n \t\tAdded tags.
\n \t\tChanged statement to make question clearer.
\n \t\t19/07/2012:
\n \t\tAdded description.
\n \t\t25/07/2012:
\n \t\tAdded tags.
\n \t\tQuestion appears to be working correctly.
\n \t\t17/08/2012:
\n \t\tMade copy to include in Simplify Algebraic Expressions exam.
\n \t\t", "description": "Express $\\log_a(x^{c}y^{d})$ in terms of $\\log_a(x)$ and $\\log_a(y)$. Find $q(x)$ such that $\\frac{f}{g}\\log_a(x)+\\log_a(rx+s)-\\log_a(x^{1/t})=\\log_a(q(x))$.
\nThere is a video included explaining the rules of logarithms by going through simplification of logs of numbers rather than algebraic expressions.
\n\n
", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}