Solutions for Algebra, Logarithms and Population Problems

This document presents solutions to several mathematical problems. The solutions cover topics including exponential equations related to population growth in West and East Goma, logarithmic equations, and simplification of logarithmic expressions. The solutions are detailed, showing each step of the process, and include references to relevant mathematical texts such as algebra and calculus books. The assignment provides a clear, step-by-step approach to solving the problems, demonstrating applications of logarithms and algebraic manipulation, offering a comprehensive guide for students studying foundational mathematics and algebra. This assignment is available on Desklib, a platform providing AI-based study tools.

1)
Answer :
Population of West Goma (in millions):
f (x) = 16.7e0.0015x
1

Population of East Goma (in millions):
g(x) = 14.6e0.0135x
The point where both the population will be equal, we have:
16.7e0.0015x= 14.6e0.0135x
16.7
14.6= e0.0135x
e0.0015x
[Property of exponents :
ea
eb = ea−b]
∴ 16.7
14.6= e0.012x
[Taking log on both sides]
ln 16.7
14.6 = 0.012x
=⇒ x = 0.1344
0.012= 11.1989
The population ofWest and East Goma willbe equalat the point x =
11.1989.Since, x = 0 corresponds to year 1998, x = 11.1989 will correspond
to year 1998 + 11.1989 = 2009.1989.
Thus the population of East and West Goma will be equal in year 2010.
2)
Answer
t = log6 36
[Take both sides as exponents to base 6]
6t = 6log6 36
[Property of exponential : aloga b = b]
∴ 6 t = 36
=⇒ 6t − 36 = 0
3)
Answer:
2

log6
3
2
[Property of logarithm : loga (c/d) = loga c − loga d]
=⇒ log6 3 − log6 2
4)
Answer :
log (x + 9) = 1 − log x
[Add log x to both sides]
log (x + 9) + log x = 1
[Property of logarithm : log (c × d) = log c + log d]
log (x(x + 9)) = 1
[Express : 1 = log 10]
log (x2 + 9x) = log 10
[Take both sides out of the log]
x2 + 9x = 10
=⇒ x 2 + 9x − 10 = 0
x2 + 10x − 1x − 10 = 0
x(x + 10) − 1(x + 10) = 0
(x + 10)(x − 1) = 0
=⇒ x = −10 or 1
The root x = −10 is not acceptable since log ofnegative numbers in not
defined.Therefore, the solution is:
x = 1
3

References :
Artin, M. (2018).Algebra.New York, New York:Pearson.
Blitzer, R. (2019).Thinking mathematically.7th ed. Boston:Pearson.
Clark, W.D. and Mccune,S.K. (2013). Linear algebra.New York, New
York: Mcgraw Hill.
Grewal,B. (2017). Advanced Engineering Mathematics.India: Mercury
Learning Information.
Grimmett,G. and Stirzaker,D. (2009).Probability and random processes.
Oxford:Oxford University Press.
Hoffman,K. and Ray Alden Kunze (2015).Linear algebra.Noida,U.P.,
India: Pearson India Education Services.
Lipschutz,S. and Lipson,M. (2018).Linear algebra.New York: Mcgraw-
Hill Education.
Maruskin, J.M. (2014).Integralcalculus.San Jose, California:Solar Crest
Publishing, Llc.
Ross,S.M. (2019).Introduction to probability models.Amsterdam [U.A.]
Academic Press.
Saxena, P. (2014).Differential calculus.New Delhi:Mcgraw Hill Education
(India) Private Limited.
Steege, R. and Bailey, K. (2018).Intermediate algebra.New York:Mcgraw-
Hill Education.
Stewart, J. (2019).Calculus:concepts and contexts.Boston, Ma, Usa:Cen-
gage.
Stewart,J., Redlin, L., Watson,S. and Panman,P. (2017). Precalculus:
mathematics for calculus.Australia:Cengage Learning.
Sullivan, M. (2019).Precalculus.Hoboken, Nj:Pearson.
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