Solving equations

Error

There was an error loading the page.

Contributors

Bill Foster

History

○

Bill Foster 11 years, 10 months ago

Created this.

There is only one version of this question that you have access to.

There is one other version that you do not have access to.

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	integer	8
b	integer	10
d	integer	1

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
random(2..5)

JME function reference

Description

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

Can an exam override the value of this variable?

Generated value: `integer`

This variable doesn't seem to be used anywhere.

Parts

Gap-fill
1. Gap Mathematical expression
  Add
1. Step Information only

Gap-fill

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

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

$x=\;$ 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 two rules for logs :

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

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

Using rule 1 we get

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

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

$\begin{eqnarray} \simplify{(x+{b})/(x+{c})}&=&{\var{a}}^{\var{d}}\Rightarrow\\ x+\var{b}&=&{\var{a}}^{\var{d}}(x+\var{c})=\simplify{{a^d}}(x+\var{c})\Rightarrow\\ \simplify{{a^d-1}x}&=&\simplify[std]{{b}-{c}*{a^d}={b-c*a^d}}\Rightarrow\\ x&=&\simplify{{b-c*a^d}/{a^d-1}} \end{eqnarray}$
We should check that this solution gives positive values for

$x+\var{b}$ and

$\simplify{x+{c}}$ as otherwise the logs are not defined.

Substituting this value for $x$ into $\log_{\var{a}}(x+\var{b})$ we get $\log_{\var{a}}(\simplify{({b-c }{a^d})/{a^d-1}})$ so OK.

For $\log_{\var{a}}(\simplify{x+{c}})$ we get on substituting for $x$ , $\log_{\var{a}}(\simplify{({b-c })/{a^d-1}})$ so OK.

Hence the value we found for $x$ is a solution to the original equation.

Give a worked solution to the whole question.

Use this tab to check that this question works as expected.

There was an error which means the tests can't run:

Part	Test	Passed?
Gap-fill
Gap-fill		Hasn't run yet
Mathematical expression
Mathematical expression		Hasn't run yet
Information only

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.
Bradford: Maths Revision. by Bill Foster in Bill's workspace.
No feedback theme by Christian Lawson-Perfect in How-tos.

Loading...

Error

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

Error

Warning

Generated value: `integer`

Parts

Gap-fill

Loading...

Error

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

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`