Katy's copy of Combining Logarithm Rules to Solve Equations

Error

There was an error loading the page.

Contributors

History

○

Katy Dobson 5 years, 6 months ago

Created this as a copy of Combining Logarithm Rules to Solve Equations.

Name	Status	Author	Last Modified
Combining Logarithm Rules to Solve Equations	Ready to use	Hannah Aldous	20/11/2019 14:34
Using Laws for Addition and Subtraction of Logarithms	Ready to use	Hannah Aldous	20/11/2019 14:44
Rearranging Logarithms involving Indices	Ready to use	Hannah Aldous	20/11/2019 14:44
Addition and Subtraction of Logarithms	Ready to use	Adrian Jannetta	16/08/2023 14:56
MATH6058 Addition and Subtraction of Logarithms	draft	Catherine Palmer	09/10/2017 11:30
Raphael's copy of Rearranging Logarithms involving Indices	draft	Raphael Stuhlmeier	27/11/2017 16:06
Addition and Subtraction of Logarithms	draft	Ida Landgärds	16/04/2018 10:43
Ida's copy of Rearranging Logarithms involving Indices	Ready to use	Ida Landgärds	09/05/2018 06:55
spørsmål 3	draft	Ida Landgärds	02/12/2020 15:42
Jonathan's copy of Using Laws for Addition and Subtraction of Logarithms	draft	Jonathan Kress	07/06/2018 01:10
Jeanne's copy of Using Laws for Addition and Subtraction of Logarithms	draft	Jeanne Parmentier	20/07/2018 09:31
Using Laws for Addition and Subtraction of Logarithms [L6 Randomised]	Needs to be tested	Matthew James Sykes	25/07/2018 11:08
Rearranging Logarithms involving Indices [L6]	draft	Abbi Mullins	24/07/2018 14:39
logarithms addition/subtraction [L6]	draft	Abbi Mullins	24/07/2018 14:49
NUMBAS-Logarithms (laws)	Ready to use	Katy Dobson	30/07/2018 10:39
David's copy of Using Laws for Addition and Subtraction of Logarithms	Ready to use	David Martin	21/09/2018 11:50
David's copy of Combining Logarithm Rules to Solve Equations	draft	David Martin	30/07/2018 14:34
David's copy of Rearranging Logarithms involving Indices	Ready to use	David Martin	17/10/2019 07:25
Joël's copy of Using Laws for Addition and Subtraction of Logarithms	draft	Jo Cohen	02/12/2020 15:42
Blathnaid's copy of spørsmål 3	draft	Blathnaid Sheridan	02/12/2020 15:42
cormac's copy of Ida's copy of Rearranging Logarithms involving Indices	draft	cormac breen	05/11/2018 13:37
cormac's copy of Ida's copy of Rearranging Logarithms involving Indices	draft	cormac breen	05/11/2018 13:38
cormac's copy of Ida's copy of Rearranging Logarithms involving Indices	draft	cormac breen	05/11/2018 13:54
Justin's copy of Using Laws for Addition and Subtraction of Logarithms	draft	Justin McGuinness	14/05/2019 14:55
Maria's copy of Rearranging Logarithms involving Indices	draft	Maria Aneiros	11/06/2019 05:32
Maria's copy of Using Laws for Addition and Subtraction of Logarithms	draft	Maria Aneiros	11/06/2019 05:30
Maria's copy of Combining Logarithm Rules to Solve Equations	draft	Maria Aneiros	25/05/2019 08:02
Using Laws for Addition and Subtraction of Logarithms	draft	Xiaodan Leng	10/07/2019 23:28
Logarithms (laws)	draft	Xiaodan Leng	10/07/2019 23:57
Using Laws for Addition and Subtraction of Logarithms	draft	Xiaodan Leng	11/07/2019 00:00
Addition and Subtraction of Logarithms	draft	Xiaodan Leng	11/07/2019 00:15
Rearranging Logarithms involving Indices	draft	Xiaodan Leng	11/07/2019 00:16
Combining Logarithm Rules to Solve Equations	draft	Xiaodan Leng	11/07/2019 01:39
Katy's copy of Combining Logarithm Rules to Solve Equations	draft	Katy Dobson	03/09/2019 08:29
Katy's copy of Using Laws for Addition and Subtraction of Logarithms	draft	Katy Dobson	03/09/2019 08:29
Musa's copy of 3 Addition and Subtraction of Logarithms	draft	Musa Mammadov	27/06/2023 01:29
Rearranging Logarithms involving Indices	Ready to use	Wan Mekwi	13/10/2020 13:19
Combining Logarithm Rules to Solve Equations	Ready to use	Wan Mekwi	13/10/2020 13:22

There are 84 other versions that do you not have access to.

Question statement

Give any introductory information the student needs.

			Name	Type	Generated Value

p	integer	4
v	integer	10
q	number	10000
m	integer	6

			Name	Type	Generated Value

x1	list	List of 8 items
y1	integer	5
z1	integer	3
b1	integer	8
c	number	512
b4	integer	3
b	number	256
b2	number	254

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
random(3..6)

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`

→ Used by:

Advice
"Part b)" - prompt
"Part b)" → "Unnamed gap" - Correct answer

Parts

Part a) Gap-fill
1. Gap Mathematical expression
  Add
Part b) 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.

Solve for $x$ .

$\log_\var{b1}(x-\var{b2})-\log_\var{b1}\left(\displaystyle\frac{1}{x}\right)=\var{b4}$

$x=$ Unnamed gap

Advice

a)

We can use the logarithm law

$k\log_a(x)=\log_a(x^k)\text{,}$

to also give a more specific rule

$\begin{align} \log_a\left(\frac{1}{x}\right)&=\log_a(x^{-1})\\ &=-\log_a(x)\text{.} \end{align}$

This means we can write our expression as

$\log_\var{b1}(x-\var{b2})+\log_\var{b1}({x})=\var{b4}\text{.}$

Then using the rule

$\log_a(x)+\log_a(y)=\log_a(x\times y)\text{,}$

we can write our equation as

$\begin{align} \log_\var{b1}(x(x-\var{b2}))&=\var{b4}\\ \log_\var{b1}(x^2-\var{b2}x)&=\var{b4}\text{.}\\ \end{align}$

We then rely on the definition of $\log_a$

$b=a^c \Longleftrightarrow \log_{a}b=c$

to write our equation as

$\begin{align} x^2-\var{b2}x&=\var{b1}^\var{b4}\\ &=\var{b1^b4}\text{.} \end{align}$

We can then write out our equation and solve either by factorising or using the quadratic formula;

$\begin{align} x^2-\var{b2}x-\var{b1^{b4}}&=0\\ (x+2)(x-\var{b})&=0\text{.} \end{align}$

As logarithms can only be applied to positive numbers, the only possible value for $x$ is $\var{b}$ .

b)

$\ln(x)$ is a shorthand for $\log_e(x)$ , so we can apply the same laws of logarithms here.

Therefore applying the rule

$k\log_a(x)=\log_a(x^k)$

we can write our equation as

$\ln(x^\var{p})+\ln(\var{q})=\var{m}\text{.}$

Then using the rule

$\log_a(x)+\log_a(y)=\log_a(x\times y)$

we can write our equation as

$\ln(\var{q}x^\var{p})=\var{m}\text{.}$

As $\ln=\log_e$ we can use

$a=b^c \Longleftrightarrow \log_ba=c$

to write our equation as

$\var{q}x^\var{p}=e^\var{m}\text{.}$

We then just need to rearrange our equation

$\begin{align} \var{q}x^\var{p}&=e^\var{m}\\[0.5em] x^\var{p}&=\frac{e^\var{m}}{\var{q}}\\[0.5em] x&=\frac{e^{\var{m}/\var{p}}}{\var{q^(1/{p})}} \end{align}$

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
Gap-fill
Gap-fill		Hasn't run yet
Mathematical expression
Mathematical expression		Hasn't run yet
Information only

This question is not used in any exams.

Loading...

Error

Katy's copy of Combining Logarithm Rules to Solve Equations

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Katy Dobson 5 years, 6 months ago

Error in variable testing condition

Error

Warning

Generated value: `integer`

Parts

Gap-fill

a)

b)

Loading...

Error

Katy's copy of Combining Logarithm Rules to Solve Equations

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Katy Dobson 5 years, 6 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

a)

b)

Generated value: `integer`