Logarithms: Solving equations 4

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Bill Foster

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Bill Foster 55 years, 3 months ago

Created this.

Name	Status	Author	Last Modified
Logarithms: Solving equations 4	draft	Bill Foster	06/08/2020 08:42
Logarithms: Solving equations 4	Ready to use	Wan Mekwi	13/10/2020 13:21

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Question statement

Find all values of $x$ that satisfy the following equations:

Give any introductory information the student needs.

			Name	Type	Generated Value

tc2	integer	7
s3	integer	-1
ans1	number	2.883
s1	integer	-1
m	integer	11
d2	integer	3
n	integer	2
a1	integer	-9
tm	integer	11
b1	integer	6
b2	integer	9
c2	integer	8
c1	integer	-2
tans1	number	2.8830397442
ans2	number	0.356
a2	integer	7
tans2	number	0.3562071871

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Data type

Value

xxxxxxxxxx
 
random(2..9)

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Description

Describe what this variable represents, and list any assumptions made about its value.

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Generated value: `integer`

This variable doesn't seem to be used anywhere.

Parts

Part a) Gap-fill
1. Gap Number entry
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Part b) Gap-fill
1. Gap Gap 0. Number entry
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2. Gap Gap 1. Number entry
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Gap-fill

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

$\var{n}^{\simplify{{a1}*x+{b1}}}=\var{m}^{\var{c1}x}$

$x=\;\;$ Unnamed gap.

Enter your answer to 3 decimal places.

Advice

Both parts of this question can be solved in a similar way, by taking logarithms of both sides of each equation.

Taking logs (to the base 10 in this case – but any base will do) of both sides of

$\var{n}^{\simplify{{a1}*x+{b1}}}=\var{m}^{\var{c1}x}$

gives on using the rule $\log(a^b)=b\log(a)$ :

$\begin{eqnarray*} \simplify[std]{({a1}x+{b1})log({n})}&=&\simplify[std]{{c1}*x*log({m})}\\ \Rightarrow\simplify[std]{x({a1}*log({n})-{c1}*log({m}))} &=&\var{-b1}\log(\var{n})\\ \Rightarrow x&=&\simplify[std]{({-b1}*log({n}))/({a1}log({n})-{c1}*log({m}))}\\ &=& \var{ans1}\mbox{ to 3 decimal places} \end{eqnarray*}$

Similarly, taking logs of both sides of:

$\var{a2}^{\var{b2}x^2}=\var{c2}^{\var{d2}x}$

gives:

$\simplify[std]{({b2}x^2)log({a2})}=\simplify[std]{{d2}*x*log({c2})} \Rightarrow \simplify[std]{x({b2}*log({a2})x-{d2}*log({c2}))} =0$

and so the solutions are:

1. $x=0$

2. $\displaystyle x=\simplify[std,!fractionNumbers]{({d2}*log({c2}))/({b2}*log({a2})) = {ans2}}$ to 3 decimal places.

Give a worked solution to the whole question.

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Part	Test	Passed?
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Number entry
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Gap-fill
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This question is used in the following exams:

Logarithms: Solving equations by Bill Foster in Bill's workspace.
Bradford: Logarithms and Solving equations by Bill Foster in Bill's workspace.
Logarithms by Gareth Woods in Gareth's workspace.
mathcentre: Logarithms and solving equations by Bill Foster in Bill's workspace.
Maths Support: Logarithms and solving equations by Christian Lawson-Perfect in Maths Support Wiki.
Kristin's copy of Maths Support: Logarithms and solving equations by Kristin Wells in Kristin's workspace.
Julie's copy of MATH6000 (WS13) Logarithms by Julie Crowley in MATH6000 Essential Mathematical Skills.
Logarthims by Aneela Arshad in Aneela's workspace.
Logarithms by Robert Pullan in Robert's workspace.
Logarithms by Deirdre Casey in Deirdre's workspace.
P8 - Exponentials & Logarithms by Rob Beckett in Rob's workspace.
Logarithms by Julie Crowley in Maths Online.
Practice: Logarithms and Solving Equations by Michael Proudman in .Algebra.
Logs by Catherine Palmer in SPIRIT Maths.
John's copy of Logarithms by John Wick in John's workspace.

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Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

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Bill Foster 55 years, 3 months ago

Error in variable testing condition

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Warning

Generated value: `integer`

Parts

Gap-fill

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Logarithms: Solving equations 4

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Bill Foster 55 years, 3 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`