Logarithms: Solving equations 2

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Created this as a copy of Logarithms: Solving equations 2.

Name	Status	Author	Last Modified
Logarithms: Solving equations 2	draft	Bill Foster	06/08/2020 08:42
Logarithms: Solving equations 2	Ready to use	Julie Crowley	26/07/2016 12:53
Regina's copy of Logarithms: Solving Equations 2	draft	Regina Kobus	07/11/2017 01:22
Regina's copy of Logarithms: Solving Equations 2	draft	Regina Kobus	07/11/2017 01:43
David's copy of Logarithms: Solving equations 2	draft	David Martin	30/07/2018 14:24
Simon's copy of Logarithms: Solving equations 2	draft	Simon Thomas	04/03/2019 11:28
Logarithms: Solving equations 2	draft	Xiaodan Leng	11/07/2019 05:47
Logarithms: Solving equations 2	Ready to use	Wan Mekwi	13/10/2020 13:20

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Question statement

Solve the following equation for $x$ .

Input your answer as a fraction or an integer as appropriate and not as a decimal.

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	2
c	number	17
b	integer	13
d	integer	1
s	integer	1
sol2	number	-15
sol1	number	-9

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Generated value: `integer`

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c
sol2

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"Unnamed part" - prompt

Parts

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1. Gap Mathematical expression
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1. Step Information only

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$2\log_{\var{a}}(\simplify{x+{b}})- \log_{\var{a}}(\simplify{(x+{c})})=\var{d}$

$x=\;$ Unnamed gap.

If you want help in solving the equation, click on Show steps. If you do so then you will lose 1 mark.

Input all numbers as fractions or integers and not as decimals.

Advice

We use the following rules for logs:

1. $\log_a(x^q)=q\log_a(x)$

2. $\log_a(\frac{x}{y})=\log_a(x)-\log_a(y)$

3. $a^x=y \iff \log_a y=x$

Using rule 1 we get

$2\log_{\var{a}}(\simplify{x+{b}})- \log_{\var{a}}(\simplify{(x+{c})})=\log_{\var{a}}((\simplify{x+{b}})^2)- \log_{\var{a}}(\simplify{(x+{c})})$
Using rule 2 gives

$\log_{\var{a}}(\simplify{(x+{b})^2})- \log_{\var{a}}(\simplify{(x+{c})})=\log_{\var{a}}\left(\simplify{(x+{b})^2/(x+{c})}\right)$
So the equation to solve becomes:

$\log_{\var{a}}\left(\simplify{(x+{b})^2/(x+{c})}\right)=\var{d}$
and using rule 3 this gives:

$\begin{eqnarray} \simplify{(x+{b})^2/(x+{c})}&=&{\var{a}}^{\var{d}}\Rightarrow\\ \simplify{(x+{b})^2}&=&{\var{a}}^{\var{d}}(\simplify{x+{c}})=\simplify{{a^d}(x+{c})}\Rightarrow\\ \simplify{x^2+{2*b-a^(d)}x+{b^2-a^(d)*c}}&=&0 \end{eqnarray}$
Solving this quadratic we get two solutions:

$x=\var{sol1}$ and $x=\var{sol2}$

We should check that these solutions gives positive values for $\simplify{x+{b}}$ and $\simplify{x+{c}}$ as otherwise the logs are not defined.

The value $x=\var{sol1}$ gives:

Substituting this value for $x$ into $\log_{\var{a}}(\simplify{x+{b}})$ we get $\log_{\var{a}}(\simplify{{2*a^d}})$ so OK.

Substituting this value for $x$ into $\log_{\var{a}}(\simplify{x+{c}})$ we get $\log_{\var{a}}(\simplify{{4*a^d}})$ so OK.

Hence $x=\var{sol1}$ is a solution to our original equation.

The value $x=\var{sol2}$ gives:

Substituting this value for $x$ into $\log_{\var{a}}(\simplify{x+{b}})$ we get $\log_{\var{a}}(\simplify{{-a^d}})$ so NOT OK.

Substituting this value for $x$ into $\log_{\var{a}}(\simplify{x+{c}})$ we get $\log_{\var{a}}(\simplify{{a^d}})$ so OK.

Hence $x=\var{sol2}$ is a solution to our original equation as $\log_{\var{a}}(\simplify{x+{b}})$ is not defined for this value of $x$ .

So there is only one solution $x=\var{sol1}$ .

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This question is used in the following exams:

Julie's copy of MATH6000 (WS13) Logarithms by Julie Crowley in MATH6000 Essential Mathematical Skills.
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.
Logs by Catherine Palmer in SPIRIT Maths.
John's copy of Logarithms by John Wick in John's workspace.

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Julie Crowley 8 years, 10 months ago

Julie Crowley 9 years, 2 months ago

Error in variable testing condition

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Generated value: `integer`

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Logarithms: Solving equations 2

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Julie Crowley 8 years, 10 months ago

Julie Crowley 9 years, 2 months ago

Error in variable testing condition

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Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`