Logarithms: Solving equations 6

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Wan Mekwi

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4 years, 5 months ago

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Wan Mekwi 4 years, 5 months ago

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Wan Mekwi 4 years, 5 months ago

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Wan Mekwi 4 years, 5 months ago

Created this as a copy of Logarithms: Solving equations 6.

Name	Status	Author	Last Modified
Logarithms: Solving equations 6	draft	Bill Foster	06/08/2020 08:42
Logarithms: Solving equations 6	Ready to use	Wan Mekwi	13/10/2020 13:19

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Question statement

Solve the following equation for $x$ . Note that there is only one solution.

$\var{c}\left(\var{a^2}^x \right)+ \var{d}\left(\var{a}^{x+1}\right)=\var{b}$

Hint: remember that $\left(A^2\right)^x=\left(A^x\right)^2$ for any number $A$ .

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	8
be	integer	-5
c	integer	4
b	integer	380
d	integer	7
ans1	number	0.774
al	integer	76
tol	integer	0
tans1	number	0.7739760316

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Generated value: `integer`

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al
tans1

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In the equation

$\var{c}\left(\var{a^2}^x \right)+ \var{d}\left(\var{a}^{x+1}\right)=\var{b}$

Let $y=\var{a}^x$ and since $\var{a^2}^x=(\var{a}^2)^x=\left(\var{a}^x\right)^2$ on substitution this becomes:

$\begin{eqnarray*} \var{c}y^2 + \var{d}\left(\var{a}y\right)&=&\var{b} \\ \\ \Rightarrow \var{c}y^2 + \var{a*d}y&=&\var{b}\Rightarrow \var{c}y^2 + \var{a*d}y-\var{b}=0\\ \\ \Rightarrow (\var{c}y+\var{al})(y-\var{abs(be)})&=&0 \mbox{ on factorisation.} \end{eqnarray*}$
This quadratic has solutions

$\displaystyle y=\simplify[std]{{-al}/{c}},\;\;y=\var{-be}$ .

But $y=\var{a}^x \gt 0$ for all $x$ and so $\displaystyle y=\simplify[std]{{-al}/{c}}$ cannot be a solution for the original equation.

We are left with $y=\var{-be}$ which gives:

$\begin{eqnarray*} \var{a}^x &=& \var{-be}\\ \Rightarrow x\ln(\var{a})&=&\ln(\var{-be})\\ \Rightarrow x&=&\frac{\ln(\var{-be})}{\ln(\var{a})} = \var{ans1} \end{eqnarray*}$

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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

Error in variable testing condition

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Generated value: `integer`

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Logarithms: Solving equations 6

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`