// Numbas version: exam_results_page_options {"name": "Simplify logarithms", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"preventleave": false, "allowregen": true, "showfrontpage": false}, "question_groups": [{"questions": [{"variablesTest": {"maxRuns": 100, "condition": ""}, "parts": [{"showCorrectAnswer": true, "marks": 0, "prompt": "\n

Express the following in terms of $\\log_a(x)$ and $\\log_a(y)$

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\$\\log_a(x^{\\var{a1}}y^{\\var{b1}})=\\alpha\\log_a(x)+\\beta\\log_a(y)\$

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$\\alpha=\\;\\;$[[0]], $\\beta=\\;\\;$[[1]]

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\$\\simplify[std]{{a2}/{b2}}\\log_a(x)+\\log_a(\\simplify{{c}*x+{d}})-\\log_a(\\simplify{x^(1/{b2})})=\\log_a(q(x))\$

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$q(x)=\\;\\;$[[0]]

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The rules for combining logs are

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\$\\begin{eqnarray*} \\log_a(bc)&=&\\log_a(b)+\\log_a(c)\\\\ \\\\ \\log_a\\left(\\frac{b}{c}\\right)&=&\\log_a(b)-\\log_a(c)\\\\ \\\\ \\log_a(b^r)&=&r\\log_a(b) \\end{eqnarray*} \$

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a)
Using these rules gives:
\$\\begin{eqnarray*} \\log_a(x^{\\var{a1}}y^{\\var{b1}})&=&\\log_a(x^{\\var{a1}})+\\log_a(y^{\\var{b1}})\\\\ &=&\\var{a1}\\log_a(x)+\\var{b1}\\log_a(y) \\end{eqnarray*} \$

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b)
\$\\begin{eqnarray*} \\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})})\\\\ \\\\ &=&\\log_a\\left(\\simplify[std]{(x^({a2}/{b2})*({c}x+{d}))/(x^(1/{b2}))}\\right)\\\\ &=&\\log_a\\left(\\simplify{x^{f}*({c}x+{d})}\\right) \\end{eqnarray*} \$

", "tags": ["checked2015", "log laws", "logarithm laws", "logarithmic expressions", "logarithms", "logs", "MAS1601", "mas1601", "rules for logarithms", "simplifying logarithms"], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "statement": "

Answer the following questions on logarithms.

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2/06/2012:

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Changed statement to make question clearer.

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19/07/2012:

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25/07/2012:

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Question appears to be working correctly.

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17/08/2012:

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Made copy to include in Simplify Algebraic Expressions exam.

", "description": "\n \t\t

Express $\\log_a(x^{c}y^{d})$ in terms of $\\log_a(x)$ and $\\log_a(y)$. Find $q(x)$ such that $\\frac{f}{g}\\log_a(x)+\\log_a(rx+s)-\\log_a(x^{1/t})=\\log_a(q(x))$

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