// Numbas version: exam_results_page_options {"name": "Graphing logarithms with vertical transformations and base>1", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"rulesets": {}, "variable_groups": [], "metadata": {"licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International", "description": "

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

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

\n

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

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Another easily found point on the curve is ${\\large(}\\var{b},$ [[0]]$\\large)$.

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$y$ increases without bound.

", "

$y$ decreases without bound.

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

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$y$ approaches $0$.

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As $x$ increases without bound,

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The vertical asymptote of $y=\\simplify{{a}log(x,{b})+{c}}$ is $x=$ [[0]].

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{graph1(1)}

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{graph1(2)}

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{graph1(3)}

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{graph1(4)}

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

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a

", "definition": "random(-3..3 except [0,1,b])", "templateType": "anything", "name": "a", "group": "Ungrouped variables"}}, "extensions": ["jsxgraph"], "type": "question", "preamble": {"js": "", "css": ""}, "advice": "

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 $x$:

\n

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

\n

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

\n

b) Substitute $x=\\var{b}$ into the equation: $y=\\simplify[!collectNumbers, basic]{{a}log({b},{b})+{c}}=\\simplify[!collectNumbers]{{a}*1+{c}}=\\var{a+c}$. Therefore, another easily found point is $(\\var{b},\\var{a+c})$.

\n

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

\n

d) An asymptote is a line or curve that approaches a given curve arbitrarily closely. For the curve $y=\\simplify{{a}log(x,{b})+{c}}$ the closer $x$ gets to zero (approaching from the right), the smaller larger $y$ gets (without bound). In other words, as $x$ approaches $0$ from the right, $y$ approaches negative infinity. This means that the asymptote for $y=\\log_{\\var{b}}(x)$ is the line $x=0$ (the $y$-axis).

\n

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

\n

{graph1(1)}

", "statement": "

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

\n

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

\n

\n

", "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}]}], "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}