Logarithms: Solving equations 3

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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 3	draft	Bill Foster	06/08/2020 08:42
Logarithms: Solving equations 3	Ready to use	Wan Mekwi	13/10/2020 13:21

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Question statement

Solve the following equation for $x$ .

$\simplify[std]{log({a1}x+{b1})-log({c1}x+{d1})={e1}}$

Give any introductory information the student needs.

			Name	Type	Generated Value

e1	integer	1
a1	integer	5
b1	integer	16
tol	integer	0
ans	number	-0.115
c1	integer	56
tans	number	-0.1153153153
d1	integer	8

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Description

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Generated value: `integer`

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Parts

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Input the solution for $x$ here:

$x=\;\;$ Unnamed gap

Input your answer to 3 decimal places.

Advice

First use one of the logarith laws which states (for logarithms to any base)

$\log(a)-\log(b)=\log\left(\frac{a}{b}\right)$

So the equation can be written as:

$\log_{10}\left(\simplify[std]{({a1}x+{b1})/({c1}x+{d1})}\right)=\var{e1}$
Now exponentiate both sides to get:

$\simplify[std,!otherNumbers]{({a1}x+{b1})/({c1}x+{d1})}=10^{\var{e1}} \Rightarrow \simplify[std,!otherNumbers]{{a1}x+{b1}=10^{e1}({c1}x+{d1})}$
Collect together terms in

$x$ on the left and everything else on the right of the equation gives:

$\simplify[std,!otherNumbers]{x({a1}-10^{e1}*{c1})=10^{e1}*{d1}-{b1}}$
Finally rearrange to get:

$\simplify[std]{x=(10^{e1}*{d1}-{b1})/({a1}-10^{e1}*{c1})={10^e1*d1-b1}/{a1-10^e1*c1}}$
which to 3 decimal places evaluates to

$x=\var{ans}.$

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This question is used in the following exams:

mathcentre: Logarithms and solving equations by Bill Foster in Bill's workspace.
Logarithms: Solving equations by Bill Foster in Bill's workspace.
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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.
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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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Metadata

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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Generated value: `integer`

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Logarithms: Solving equations 3

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`