// Numbas version: exam_results_page_options {"name": "Jamie's copy of Graphing logarithms with horizontal and vertical transformations and base>1 ", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "statement": "

The following questions will gauge your understanding of logarithms and how to graph them.

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The logarithm you will be working with for this question is \$y=\\simplify{{a}log({s}x+{d},{b})+{c}}.\$

This question assumes you understand the definition and the laws of logarithms.

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a) To find the $x$-intercept, substitute $y=0$ into the equation and solve for $x$:

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\\begin{align*}&&0&=\\simplify{{a}log({s}x+{d},{b})+{c}}\\\\\\implies&& \\var{-c}&=\\simplify{{a}log({s}x+{d},{b})}\\\\\\implies&&\\simplify[fractionNumbers,simplifyFractions,basic]{-{c/a}}&=\\simplify{log({s}x+{d},{b})}\\\\\\implies&&\\simplify[fractionNumbers,simplifyFractions,basic]{{b}^(-{c/a})}&=\\simplify{{s}x+{d}}\\\\\\implies&&\\simplify[fractionNumbers,simplifyFractions,basic,unitDenominator,unitFactor]{({b}^(-{c/a})-{d})/{s}}&=x\\end{align*}

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Therefore, the $x$-intercept is the point $\\left(\\simplify[fractionNumbers,simplifyFractions,basic,unitDenominator,unitFactor]{({b}^(-{c/a})-{d})/{s}},0\\right)$.

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b) Substitute $x=\\simplify[fractionNumbers]{{(b-d)/s}}$ into the equation: $y=\\simplify[!collectNumbers,basic,fractionNumbers]{{a}log(({(b-d)})+{d},{b})+{c}}=\\simplify[!collectNumbers,basic]{{a}log({b},{b})+{c}}=\\var{a+c}$. Therefore, another easily found point is $\\left(\\simplify[fractionNumbers]{{(b-d)/s}},\\var{a+c}\\right)$.

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c) As $x$ gets larger and larger (increases without bound, or approaches infinity) $y=\\simplify{{a}log({s}x+{d},{b})+{c}}$ gets smaller and smaller (decreases without bound, or approaches negative infinity). larger and larger (increases without bound, or approaches infinity).

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c) As $x$ gets smaller and smaller (decreases without bound, or approaches negative infinity) $y=\\simplify{{a}log({s}x+{d},{b})+{c}}$ gets smaller and smaller (decreases without bound, or approaches negative infinity). larger and larger (increases without bound, or approaches infinity).

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d) An asymptote is a line or curve that approaches a given curve arbitrarily closely. For the curve $y=\\simplify{{a}log({s}x+{d},{b})+{c}}$ the closer $\\simplify{{s}x+{d}}$ gets to zero (approaching from the right), the smaller larger $y$ gets (without bound). In other words, as $x$ approaches $\\simplify[fractionNumbers]{{-d/s}}$ from the right left, $y$ approaches negative infinity. This means that the asymptote for $y=\\simplify{{a}log({s}x+{d},{b})+{c}}$ is the line $x=\\simplify[fractionNumbers]{{-d/s}}$.

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e) Given all the information above, it should be clear that the graph should look like

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{graph1(1)}

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a

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Graphing $y=a\\log_{b}(\\pm x+d)+c$

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The $x$-intercept of $y=\\simplify{{a}log({s}x+{d},{b})+{c}}$ is the point $\\large($[[0]], [[1]]$\\large)$.

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Note: You should input the exact answer, not just a decimal approximation.

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Another important point on the curve is ${\\large(}\\simplify[fractionNumbers]{{(b-d)/s}},$ [[0]]$\\large)$.

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As $x$ increases decreases without bound,

", "choices": ["

$y$ increases without bound.

", "

$y$ decreases without bound.

", "

$y$ approaches $\\var{c}$.

", "

$y$ approaches $0$.

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Which graph best represents $y=\\simplify{{a}log({s}x+{d},{b})+{c}}$?

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{graph1(1)}

", "

{graph1(2)}

", "

{graph1(3)}

", "

{graph1(4)}

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