Logarithms: Solving equations 4

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Wan Mekwi

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Wan Mekwi 4 years, 5 months ago

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Created this as a copy of Logarithms: Solving equations 4.

Name	Status	Author	Last Modified
Logarithms: Solving equations 4	draft	Bill Foster	06/08/2020 08:42
Logarithms: Solving equations 4	Ready to use	Wan Mekwi	13/10/2020 13:21

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Question statement

Find all values of $x$ that satisfy the following equations:

Give any introductory information the student needs.

			Name	Type	Generated Value

tc2	integer	8
s3	integer	-1
ans1	number	-0.296
s1	integer	1
m	integer	13
d2	integer	5
n	integer	7
a1	integer	9
tm	integer	13
b1	integer	5
b2	integer	6
c2	integer	8
c1	integer	-6
tans1	number	-0.295705073
ans2	number	1.250
a2	integer	4
tans2	number	1.25

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Description

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Generated value: `integer`

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Parts

Part a) Gap-fill
1. Gap Number entry
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Part b) Gap-fill
1. Gap Gap 0. Number entry
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2. Gap Gap 1. Number entry
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Ask the student a question, and give any hints about how they should answer this part.

$\var{n}^{\simplify{{a1}*x+{b1}}}=\var{m}^{\var{c1}x}$

$x=\;\;$ Unnamed gap.

Enter your answer to 3 decimal places.

Advice

Both parts of this question can be solved in a similar way, by taking logarithms of both sides of each equation.

Taking logs (to the base 10 in this case – but any base will do) of both sides of

$\var{n}^{\simplify{{a1}*x+{b1}}}=\var{m}^{\var{c1}x}$

gives on using the rule $\log(a^b)=b\log(a)$ :

$\begin{eqnarray*} \simplify[std]{({a1}x+{b1})log({n})}&=&\simplify[std]{{c1}*x*log({m})}\\ \Rightarrow\simplify[std]{x({a1}*log({n})-{c1}*log({m}))} &=&\var{-b1}\log(\var{n})\\ \Rightarrow x&=&\simplify[std]{({-b1}*log({n}))/({a1}log({n})-{c1}*log({m}))}\\ &=& \var{ans1}\mbox{ to 3 decimal places} \end{eqnarray*}$

Similarly, taking logs of both sides of:

$\var{a2}^{\var{b2}x^2}=\var{c2}^{\var{d2}x}$

gives:

$\simplify[std]{({b2}x^2)log({a2})}=\simplify[std]{{d2}*x*log({c2})} \Rightarrow \simplify[std]{x({b2}*log({a2})x-{d2}*log({c2}))} =0$

and so the solutions are:

1. $x=0$

2. $\displaystyle x=\simplify[std,!fractionNumbers]{({d2}*log({c2}))/({b2}*log({a2})) = {ans2}}$ to 3 decimal places.

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This question is used in the following exam:

Exponentials and Logarithms by Wan Mekwi in Introduction to Calculus.

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Wan Mekwi 4 years, 5 months ago

Wan Mekwi 4 years, 5 months ago

Wan Mekwi 4 years, 5 months ago

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Generated value: `integer`

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Logarithms: Solving equations 4

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Wan Mekwi 4 years, 5 months ago

Wan Mekwi 4 years, 5 months ago

Wan Mekwi 4 years, 5 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`