// Numbas version: exam_results_page_options {"name": "Logarithms (laws)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"rulesets": {}, "variable_groups": [], "tags": [], "name": "Logarithms (laws)", "functions": {}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

Use laws for addition and subtraction of logarithms to simplify a given logarithmic expression to an arbitrary base.

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When adding and subtracting logarithms we can simplify the expressions using some logarithm laws. These laws are

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\\\begin{align} \\log_a(x)+\\log_a(y)&=\\log_a(xy)\\text{,}\\\\ \\log_a(x)-\\log_a(y)&=\\log_a\\left(\\frac{x}{y}\\right)\\text{.} \\end{align}\

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$\\log_a(\\var{x1[1]})+ \\log_a(\\var{x1[0]})=\\log_a($ [[0]]$)$

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$\\log_a(\\var{(x1[4])*y1})-\\log_a(\\var{x1[4]})=\\log_a($ [[0]]$)$

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$\\log_a(\\var{y2})+\\log_a(\\var{(x1[4])*y3})-\\log_a(\\var{x1[4]})=\\log_a($[[0]]$)$

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random(2..6)

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#### a)

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We need to use the rule

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\$\\log_a(x)+\\log_a(y)=\\log_a(xy)\\text{.}\$

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Substituting in our values for $x$ and $y$ gives

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\\\begin{align} \\log_a(\\var{x1[1]})+\\log_a(\\var{x1[0]})&=\\log_a(\\var{x1[1]}\\times \\var{x1[0]})\\\\ &=\\log_a(\\var{x1[1]*x1[0]})\\text{.} \\end{align}\

\n

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#### b)

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We need to use the rule

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\$\\log_a(x)-\\log_a(y)=\\log_a\\left(\\frac{x}{y}\\right)\\text{.}\$

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Substituting in our values for $x$ and $y$ gives

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\\\begin{align} \\log_a(\\var{x1[4]*y1})-\\log_a(\\var{x1[4]})&=\\log_a(\\var{x1[4]*y1}\\div \\var{x1[4]})\\\\ &=\\log_a(\\var{y1})\\text{.} \\end{align}\

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Simplify the expressions to fill in the gaps.

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