Logarithms: Solving Logarithmic Equations 7 - Sum of Logarithms

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Question statement

Solve for $x$ :

$\log(y)+\log(x)=\frac{1}{\var{n}}\log(\var{a^n}y^\var{n}).$

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			Name	Type	Generated Value

			a	integer	6
			n	integer	2

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Advice

To solve for $x$ , we want to make use of the following logarithm rules:

$\log(a)+\log(b) = \log(ab)$ ;
$\log(a^n)=n \log(a)$ .

Therefore,

$\begin{split} &\log(y)+&\log(x) &\,= \frac{1}{\var{n}} \log(\var{a^n}y^\var{n}) \\ &\implies &\log(yx) &\,= \log\left((\var{a^n}y^\var{n})^\frac{1}{\var{n}})\right) \\ &\implies &\log(yx) &\,= \log(\var{a}y) \end{split}$

Recall that if $\log(a) = \log(b)$ , then $a=b$ .

Hence,

$\begin{split} \log(yx) &\,= \log(\var{a}y) \\ \implies \quad yx &\,= \var{a}y \\ \implies \quad x &\,= \var{a}. \end{split}$

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This question is used in the following exams:

Logarithms and Exponentials - Calculating Unknowns by Ben McGovern in MASH Bath: Moodle quizzes and TS.
Practice on logarithm by Jinhua Mathias in Maths 3-2020.
All the good stuff by Chris Templet in Chris's workspace.
MathRevise by Nina Strydom in Nina's workspace.
Summer Quiz 2 by Kate Vincent in Kate's workspace.
EGD125_Quiz1 by Matt Falk in Matt's workspace.
MATH2411 - Logarithms by Blathnaid Sheridan in Blathnaid's workspace.
MATH2411 - Assessment 1 by Blathnaid Sheridan in Blathnaid's workspace.

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Logarithms: Solving Logarithmic Equations 7 - Sum of Logarithms

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Christian Lawson-Perfect 2 years, 7 months ago

Ben McGovern 3 years, 8 months ago

Ben McGovern 3 years, 10 months ago

Error in variable testing condition

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Generated value: `integer`

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Logarithms: Solving Logarithmic Equations 7 - Sum of Logarithms

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Christian Lawson-Perfect 2 years, 7 months ago

Ben McGovern 3 years, 8 months ago

Ben McGovern 3 years, 10 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`