Solving equations

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Bill Foster

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Bill Foster 11 years, 10 months ago

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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	11
b	integer	3
d	integer	2
s	integer	1
sol2	number	-7
sol1	number	5

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sol2

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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 three rules for logs :

1. $n\log_a(m)=\log_a(m^n)$

2. $\log_a(b)-\log_a(c)=\log_a(b/c)$

3. $\log_a(p)=r \Rightarrow p=a^r$

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:

Maths Revision. by Bill Foster in Bill's workspace.
Exponentials, logarithms and functions by Peter Chapman in Peter's workspace.
Robert's copy of Exponentials, logarithms and functions by Robert Pullan in Robert's workspace.
Algebra: 1 by Aneela Arshad in Aneela's workspace.
Bradford: Maths Revision. by Bill Foster in Bill's workspace.
Algebraic Fractions 1 by Aneela Arshad in Aneela's workspace.
No feedback theme by Christian Lawson-Perfect in How-tos.

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Taxonomy: mathcentre

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Bill Foster 11 years, 10 months ago

Error in variable testing condition

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Generated value: `integer`

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Solving equations

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Bill Foster 11 years, 10 months ago

Error in variable testing condition

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Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`