Solving log equations using exponentials

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Adrian Jannetta 7 years, 6 months ago

Created this as a copy of Solving log equations using exponentials.

Name	Status	Author	Last Modified
Solving exponential equations using logs	Ready to use	Ben Brawn	17/12/2021 21:52
Solving log equations using exponentials	Ready to use	Ben Brawn	25/10/2017 10:55
Solving exponential equations using logs	draft	steve kilgallon	04/08/2016 11:39
Solving log equations using exponentials	draft	steve kilgallon	04/08/2016 11:39
Solving log equations using exponentials	draft	Adrian Jannetta	16/08/2023 14:56
Bernard's copy of Solving exponential equations using logs	draft	Bernard Enright	16/01/2018 13:37
David's copy of Solving exponential equations using logs	draft	David Martin	30/07/2018 14:30
David's copy of Solving log equations using exponentials	draft	David Martin	30/07/2018 14:30
cormac's copy of Solving log equations using exponentials	draft	cormac breen	30/10/2018 15:51
Solving exponential equations using logs	draft	Xiaodan Leng	11/07/2019 01:06
Solving log equations using exponentials	draft	Xiaodan Leng	11/07/2019 01:07

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Question statement

Solve the following equation for $x$ .

Give any introductory information the student needs.

			Name	Type	Generated Value

			Name	Type	Generated Value

power	integer	-1
d	integer	9
f	integer	4
g	integer	-5
b	integer	5
a	integer	11
c	integer	4
ans	decimal	dec("-3.454545454545454545454545454545454545455e-1")

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Generated value: `integer`

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1. Gap Mathematical expression
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$\displaystyle{\var{d}\log_\var{b}(\simplify{{a}x+{c}})+\var{f}=\var{g} }.$

$x=$ Unnamed gap

Note: Please give your answer as a fraction if an exact value cannot be found.

Advice

We start solving the equation one operation at a time by doing the inverse to both sides, when we get to undoing the log we recall the definition of $\log_b$ , write the equation in index form and continue solving.

Recall: The definition of $\log_b$ says $\log_b(a)=c$ is equivalent to $b^c=a$ .

$\displaystyle{\var{d}\log_\var{b}(\simplify{{a}x+{c}})+\var{f} }$	$=$	$\var{g}$
$\displaystyle{\var{d}\log_\var{b}(\simplify{{a}x+{c}}) }$	$=$	$\var{g-f}$	(subtract $\var{f}$ from both sides)
$\displaystyle{\log_\var{b}(\simplify{{a}x+{c}}) }$	$=$	$\var{power}$	(divide both sides by $\var{d}$ )
$\simplify[basic]{{b}^{power}}$	$=$	$\simplify{{a}x+{c}}$	(using the definition of $\log_\var{b}$ )
$\simplify[basic,unitpower]{{b}^{power}-{c}}$	$=$	$\var{a}x$	(subtract $\var{c}$ from both sides)

$\displaystyle{\simplify[basic,fractionnumbers,unitpower]{({b}^{power}-{c})/{a}}}$	$=$	$x$	(divide both sides by $\var{a}$ )

$x$	$=$	$\displaystyle{\simplify[basic,fractionnumbers,unitpower]{{({b}^{power}-{c})/{a}}}}$

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Adrian Jannetta 7 years, 6 months ago

Error in variable testing condition

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Generated value: `integer`

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Solving log equations using exponentials

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Adrian Jannetta 7 years, 6 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`