Combining Logarithm Rules to Solve Equations

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From users who are members of Transition to university :

Christian Lawson-Perfect	said	Ready to use	7 years, 8 months ago
Hannah Aldous	said	Needs to be tested	7 years, 8 months ago
Elliott Fletcher	said	Has some problems	7 years, 8 months ago
Stanislav Duris	said	Has some problems	7 years, 9 months ago

History

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Elliott Fletcher 7 years, 8 months ago

Published this.

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Christian Lawson-Perfect 7 years, 8 months ago

Gave some feedback: Ready to use

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Christian Lawson-Perfect 7 years, 8 months ago

Saved a checkpoint:

This is fine. I had to add \left and \right around brackets in a couple of places so they stretch around the fractions inside.

Restore this checkpoint

○

Hannah Aldous 7 years, 8 months ago

Gave some feedback: Needs to be tested

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Elliott Fletcher commented 7 years, 8 months ago

This is a really good question, i just have a couple of comments

Main Parts

a) and b) i would use a \displaystyle before the fractions in the questions to make the fractions look bigger.

Advice

a) in the very last calculation when you are factorising the quadratic, you write the constant term as a power of a different number like 4^3, i would just leave this as it's simplified number, e.g

x^2-30x-4^3 = 0 could be x^2-30x-64 = 0

b) There is a random ' after the ln on the first line here

I think you are just missing a few full stops after some equations. Specifically, the one before "We then just need to rearrange our equation" and the very last line where you give the final value of x.

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Elliott Fletcher 7 years, 8 months ago

Gave some feedback: Has some problems

○

Hannah Aldous 7 years, 9 months ago

Gave some feedback: Needs to be tested

○

Hannah Aldous 7 years, 9 months ago

Gave some feedback: Has some problems

○

Hannah Aldous 7 years, 9 months ago

Gave some feedback: Doesn't work

○

Hannah Aldous 7 years, 9 months ago

Gave some feedback: Needs to be tested

○

Stanislav Duris 7 years, 9 months ago

Gave some feedback: Has some problems

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Stanislav Duris commented 7 years, 9 months ago

I feel like some parts in advice need a bit more explanation:

- In part a.iv) I think you should add a bit more explanation why the answer is always 0. Something along the lines that as $b^0 = 1$ then $\log_b(1) = 0$ .

- In part b) at the end, "the only value for x is..." I feel should be something along the lines of "the only possible value for x is ...".

- In part c) "Laws for logarithms can also be applied to $\ln$ ." maybe you could add "as $\ln(x) = \log_e(x)$ ."

- Also, the last two lines of the advice is only right when p = 2. The right answer in advice always takes a square root rather than the appropriate root. Fix this.

There are some small mistakes in punctuation:

- Some parts are missing a blank space between the = and the gap.

- In advice, I feel like many of the commas you used are unnecesseary, for example "We need to use the rule," is followed by that rule so I would remove the comma so it flows better. You've done this correctly in part a.iii). Maybe if you want to keep the comma there, you should put comma after the rule/equation on the separate line as well, to make it stand out from the sentence. Similarly, "Subsituting in our values for x and y gives," I would remove these commas too for the same reason.

- You're missing a few full stops when your sentences finish with an equation in advice. For example, in part a) after the last line in each part i/ii/iii or last line of the whole advice, I think there should be full stops.

- In advice part b), near the end - "to write our equation as," is followed by two lines of equations. You put full stop after the first line by mistake, but it should be at the end of the second line. Just move the \text{.} to the end of that expression.

Finally, part c) asks for the answer to be put in the form $\frac{e^{a}}{b}$ , but accepts answers in different forms such as ( $(\frac{e^{m}}{q})^{1/p}$ or ${e^{a}}*{b^{-1}}$ . I think this could be easily fixed by putting some required/forbidden strings in the string restriction for the question. I would advise trying "(e" and ")^" and "*" as forbidden strings and add "e" as a required string just to be sure. Hopefully that will fix this tiny problem.

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Hannah Aldous 7 years, 9 months ago

Gave some feedback: Needs to be tested

○

Hannah Aldous 7 years, 9 months ago

Created this.

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

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			Name	Type	Generated Value

p	integer	6
v	integer	4
q	number	4096
m	integer	10

			Name	Type	Generated Value

x1	list	List of 8 items
y1	integer	5
z1	integer	3
b1	integer	6
c	number	36
b4	integer	2
b	number	18
b2	number	16

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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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Combining Logarithm Rules to Solve Equations

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Elliott Fletcher 7 years, 8 months ago

Christian Lawson-Perfect 7 years, 8 months ago

Christian Lawson-Perfect 7 years, 8 months ago

Hannah Aldous 7 years, 8 months ago

Elliott Fletcher commented 7 years, 8 months ago

Elliott Fletcher 7 years, 8 months ago

Hannah Aldous 7 years, 9 months ago

Hannah Aldous 7 years, 9 months ago

Hannah Aldous 7 years, 9 months ago

Hannah Aldous 7 years, 9 months ago

Stanislav Duris 7 years, 9 months ago

Stanislav Duris commented 7 years, 9 months ago

Hannah Aldous 7 years, 9 months ago

Hannah Aldous 7 years, 9 months ago

Error in variable testing condition

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Generated value: `integer`

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b)

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Combining Logarithm Rules to Solve Equations

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Elliott Fletcher 7 years, 8 months ago

Christian Lawson-Perfect 7 years, 8 months ago

Christian Lawson-Perfect 7 years, 8 months ago

Hannah Aldous 7 years, 8 months ago

Elliott Fletcher commented 7 years, 8 months ago

Elliott Fletcher 7 years, 8 months ago

Hannah Aldous 7 years, 9 months ago

Hannah Aldous 7 years, 9 months ago

Hannah Aldous 7 years, 9 months ago

Hannah Aldous 7 years, 9 months ago

Stanislav Duris 7 years, 9 months ago

Stanislav Duris commented 7 years, 9 months ago

Hannah Aldous 7 years, 9 months ago

Hannah Aldous 7 years, 9 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

a)

b)

Generated value: `integer`