Solving exponential equations using logs

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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

Give any introductory information the student needs.

			Name	Type	Generated Value

FV	integer	44500
pay	integer	200
int	number	0.05
periods	number	51.1428784736
test	number	11.125

			Name	Type	Generated Value

a	integer	1380
p	integer	2
c	integer	100
b	number	1.25
frac	rational	640
logfrac	decimal	dec("2.806179973983887171282433368346958160609")
logb	number	0.096910013
d	integer	12
amc	integer	1280
n	number	347.4786417066

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Generated value: `integer`

44500

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periods
test

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Parts

Part a) Gap-fill
1. Gap Mathematical expression
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Part b) Gap-fill
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Solve the following equation for $n$

$\begin{align*}\simplify{{a}={p}({b})^(n/{d})+{c}}.\end{align*}$

$n=$ Unnamed gap

Note: Typing $\log(5)$ will input the value $\log_{10}(5)$ , whereas $\log5$ will not work.
Note: Typing $\ln(5)$ will input the value $\log_e(5)$ , whereas $\ln5$ will not work.

Advice

We start solving the equation one operation at a time by doing the inverse to both sides, when we get to undoing the exponential we apply a log to both sides, we then use a log law and continue solving.

$\var{a}$	$=$	$\simplify{{p}({b})^(n/{d})+{c}}$
$\simplify{{a-c}}$	$=$	$\simplify{{p}({b})^(n/{d})}$	(subtract $\var{c}$ from both sides)
$\var{frac}$	$=$	$\simplify{{b}^(n/{d})}$	(divide both sides by $\var{p}$ )
$\log(\var{frac})$	$=$	$\log(\var{b}^{\frac{n}{\var{d}}})$	(take the log of both sides)
	$=$	$\frac{n}{\var{d}}\log(\var{b})$	(use a log law)

$\displaystyle{\frac{\log(\var{frac})}{\log(\var{b})}}$	$=$	$\frac{n}{\var{d}}$	(divide both sides by $\log(\var{b})$ )

$\displaystyle{\frac{\var{d}\log(\var{frac})}{\log(\var{b})}}$	$=$	$n$	(multiply both sides by $\var{d}$ )

${\var{FV}}$	$=$	$\displaystyle{\simplify{{pay}((1+{int})^n-1)/{int}}}$
$\simplify{{FV*int}}$	$=$	$\displaystyle{\simplify{{pay}((1+{int})^n-1)}}$	(multiply both sides by $\var{int}$ )
$\displaystyle{\simplify[simplifyFractions]{{FV*int}/{pay}}}$	$=$	$\displaystyle{\simplify{(1+{int})^n-1}}$	(divide both sides by $\var{pay}$ )
$\displaystyle{\simplify[simplifyFractions]{{FV*int}/{pay}+1}}$	$=$	$\displaystyle{\simplify{(1+{int})^n}}$	(add $1$ to both sides)
$\displaystyle{\simplify[simplifyFractions]{{FV*int+pay}/{pay}}}$	$=$	$\displaystyle{\simplify{(1+{int})^n}}$	(tidy up left hand side)

$\displaystyle{\log\left(\simplify[simplifyFractions]{{FV*int+pay}/{pay}}\right)}$	$=$	$\displaystyle{\log\left((\var{1+int})^n\right)}$	(take the log of both sides)

$\displaystyle{\log\left(\simplify[simplifyFractions]{{FV*int+pay}/{pay}}\right)}$	$=$	$\displaystyle{n\log(\var{1+int})}$	(use a log law)

$\displaystyle{\frac{\log\left(\simplify[simplifyFractions]{{FV*int+pay}/{pay}}\right)}{\log(\var{1+int})}}$	$=$	$n$	(divide both sides by $\log(\var{1+int})$ )

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This question is used in the following exams:

Solving equations involving logs and exponentials by Ben Brawn in Algebra.
Solving exponential equations - ACFI by Ben Brawn in Algebra.
NUMBAS - Solving equations by Katy Dobson in Katy's workspace.
International Summer School: Logs and Exponentials 2 by Rachel Binks in MSP International Summer School.
Solving equations by using logs by Ben Brawn in Algebra.
Algebra by Dean Ang in Dean's workspace.
Algebra by Chris Templet in Chris's workspace.
Core Maths Skills Final Exam Practice Questions by Maria Aneiros in MY QUESTIONS.

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Solving exponential equations using logs

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Taxonomy: mathcentre

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Ben Brawn 9 years, 1 month ago

Ben Brawn 9 years, 1 month ago

Error in variable testing condition

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Generated value: integer

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Generated value: `integer`