Maria's copy of Combining Logarithm Rules to Solve Equations

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Maria Aneiros 5 years, 10 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

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			Name	Type	Generated Value

p	integer	6
v	integer	4
q	number	4096
m	integer	2

			Name	Type	Generated Value

x1	list	[ 7, 19, 19, 18, 20, 8, 7, 5 ]
y1	integer	4
z1	integer	3
b1	integer	6
c	number	36
b4	integer	2
b	number	18
b2	number	16

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Generated value: `integer`

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

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1. Gap Mathematical expression
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Part b) Gap-fill
1. Gap Mathematical expression
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1. Step Information only

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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

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

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Maria's copy of Combining Logarithm Rules to Solve Equations

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Taxonomy: mathcentre

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Maria Aneiros 5 years, 10 months ago

Error in variable testing condition

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Warning

Generated value: `integer`

Parts

Gap-fill

a)

b)

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Maria's copy of Combining Logarithm Rules to Solve Equations

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Maria Aneiros 5 years, 10 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

a)

b)

Generated value: `integer`