Graphing logarithms with horizontal and vertical transformations and base>1

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Ben Brawn

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Ben Brawn

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7 years, 8 months ago

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Ben Brawn 7 years, 8 months ago

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Ben Brawn 7 years, 8 months ago

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Ben Brawn 7 years, 8 months ago

Created this as a copy of Graphing exponentials with horizontal and vertical transformations and base>1 .

Name	Status	Author	Last Modified
Graphing exponentials of the form y=b^x with b>1	Ready to use	Ben Brawn	07/03/2025 05:47
Graphing exponentials of the form y=b^x with 0<b<1	Ready to use	Ben Brawn	07/03/2025 05:47
Graphing exponentials with vertical transformations and base>1	Ready to use	Ben Brawn	07/03/2025 05:47
Graphing exponentials with horizontal and vertical transformations and base>1	Ready to use	Ben Brawn	07/03/2025 05:47
Graphing logarithms of the form y=log_{b}(x) with b>1	Ready to use	Ben Brawn	03/03/2025 00:25
Graphing logarithms with vertical transformations and base>1	Ready to use	Ben Brawn	03/03/2025 00:25
Graphing logarithms with horizontal and vertical transformations and base>1	Ready to use	Ben Brawn	03/03/2025 00:25
Jamie's copy of Graphing logarithms of the form y=log_{b}(x) with b>1	draft	Jamie Wood	19/09/2017 12:36
Jamie's copy of Graphing logarithms with vertical transformations and base>1	draft	Jamie Wood	19/09/2017 12:38
Jamie's copy of Graphing exponentials with horizontal and vertical transformations and base>1	draft	Jamie Wood	19/09/2017 13:33
Jamie's copy of Graphing exponentials with vertical transformations and base>1	draft	Jamie Wood	19/09/2017 13:29
Jamie's copy of Graphing logarithms with horizontal and vertical transformations and base>1	draft	Jamie Wood	19/09/2017 12:43
Jamie's copy of Graphing exponentials of the form y=b^x with b>1	draft	Jamie Wood	19/09/2017 13:07
Maria's copy of Graphing exponentials of the form y=b^x with b>1	draft	Maria Aneiros	25/05/2019 08:02
Exam04 - Exponentials	draft	Michael Proudman	07/12/2021 09:54
nuExam04 - Exponentials	draft	Michael Proudman	07/12/2021 09:54
Graphing exponentials with vertical transformations and base>1	draft	Xiaodan Leng	11/07/2019 01:13
Graphing exponentials with vertical transformations and base>1	draft	Xiaodan Leng	11/07/2019 01:26
Graphing exponentials with horizontal and vertical transformations and base>1	draft	Xiaodan Leng	11/07/2019 01:27
Graphing exponentials of the form y=b^x with 0<b<1	draft	Xiaodan Leng	11/07/2019 01:31
Graphing exponentials of the form y=b^x with b>1	draft	Xiaodan Leng	11/07/2019 01:32
Graphing logarithms with horizontal and vertical transformations and base>1	draft	Xiaodan Leng	11/07/2019 02:17
Graphing logarithms of the form y=log_{b}(x) with b>1	draft	Xiaodan Leng	11/07/2019 02:18
Graphing logarithms with vertical transformations and base>1	draft	Xiaodan Leng	11/07/2019 02:18

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Question statement

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

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

Give any introductory information the student needs.

			Name	Type	Generated Value

b	integer	2
a	integer	3
c	integer	-5
d	integer	-3
s	integer	-1

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Value

xxxxxxxxxx
 
random(2..6)

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Description

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Generated value: `integer`

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Parts

Part a) Gap-fill
1. Gap Gap 0. Mathematical expression
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2. Gap Gap 1. Mathematical expression
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Part b) Gap-fill
1. Gap Number entry
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Part c) Choose one from a list
Part d) Gap-fill
1. Gap Number entry
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Part e) Choose one from a list

Gap-fill

Ask the student a question, and give any hints about how they should answer this part.

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

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

Advice

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

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

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

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

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 $\left(\simplify[fractionNumbers]{{(b-d)/s}},\var{a+c}\right)$.

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).

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).

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 $x=\simplify[fractionNumbers]{{-d/s}}$.

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

{graph1(1)}

Give a worked solution to the whole question.

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Part	Test	Passed?
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Mathematical expression
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Gap-fill
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Number entry
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Choose one from a list
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Gap-fill
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Number entry
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Choose one from a list
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This question is used in the following exams:

Graphing logarithms by Ben Brawn in Graphing and Polynomials.
Graphing and Polynomials by Chris Templet in Chris's workspace.

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Graphing logarithms with horizontal and vertical transformations and base>1

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Taxonomy: mathcentre

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Taxonomy: Context

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Ben Brawn 7 years, 8 months ago

Ben Brawn 7 years, 8 months ago

Ben Brawn 7 years, 8 months ago

Error in variable testing condition

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Generated value: `integer`

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Gap-fill

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Error

Graphing logarithms with horizontal and vertical transformations and base>1

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Ben Brawn 7 years, 8 months ago

Ben Brawn 7 years, 8 months ago

Ben Brawn 7 years, 8 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`