UNIT 2:LOGARITHMIC AND EXPONENTIAL EQUATIONS
Key unit competence: Solve equations involving logarithms or exponentialsand apply them to model and solve related problems.
2.0 INTRODUCTORY ACTIVITY
2.3 Application of exponential and logarithmic equations in real life
2.3.1 Application of exponential equations to estimate thePopulation Growth
A population whose rate of decrease is proportional to the size of the population
at any time obeys a law of the forms P = Ae−kt . The negative sign on exponent
indicates that the population is decreasing. This is known as exponential
decay.
If a quantity has an exponential growth model, then the time required for it
to double in size is called the doubling time. Similarly, if a quantity has an
exponential decay model, then the time required for it to reduce in value by
half is called the halving time. For radioactive elements, halving time is calledhalf-life.
2.3.3 Application of logarithmic equations to determine the
magnitude of an earthquake
ACTIVITY 2.3.3
Using internet and books, carry out a research and find out
how logarithms can intervene to solve problems related to thedetermination of the magnitude of an earthquake.
APPLICATION ACTIVITY 2.3.3
Earth quake can occur in any country. Assuming that there was an
earthquake I*which occurred in Rwanda in a certain past year.
Discuss how we can measure eventual earthquake which may occur
in our country referring to I* instead of referring to earthquake thathappened in western countries.
2.3.4 Application of exponential equations on interest rate problems
ACTIVITY 2.3.4
Mr Cauchy has a rentable house for which he asked 20,000Frw at
the first month. However, the client has pay at the beginning of every
month by adding 1% of the money paid for the previous month. If
the money is to be paid at the new bank account for Cauchy,
a) Calculate the money kept on Cauchy’s account in the middle of
the second, the third and the fourth month.
b) What is the type of sequence made by the money to be paid by
Cauchy’s client? Determine its general term.
c) Discuss the formula to be used to find the money Mr Cauchywill find on his account at the end of 12 months.
Example
1) Mr. John operates an account with a certain bank which pays a compound
interest rate of 13.5% per annum. He opened the account at the beginning of
the year with 500,000 FRW and deposits the same amount of money at the
beginning of every year. Calculate how much he will receive at the end of 9
years.
After how long will the money have accumulated to 3.32 million of RwandanFracs?
Hence it will take 4.6 years for the amount to accumulate to 3.32 million FRW
2) A man deposits 800,000 FRW into his savings account on which interest is
15% per annum. If he makes no withdrawals, after how many years will hisbalance exceed 8 million FRW?
APPLICATION ACTIVITY 2.3.4
What annual rate of interest compounded annually should you seekif you want to double your investment in 5 years?
2.3.5 Application of exponential equations to determine the
mortgage payments
ACTIVITY 2.3.5
A loan with a fixed rate of interest is said to be amortized if both
principal and interest are paid by a sequence of equal paymentsmade over equal periods of time.
Examples:
1) Mr. Clement has just purchased a radio of 300,000Frw and has made a down
payment of 60,000Frw. He can amortize the balance (300,000Frw-60,000Frw)
at 6% for 30 years.
(a) What are the monthly payments?
(b) What is his total interest payment?
(c) After 20 years, what equity does he have in his radio (that is, what is thesum of the down payment and the amount paid on the loan)?
2) When Mr. Thomas Rwambikana died, he left an inheritance of 15,000Frw
for his family to be paid to them over a 10-year period in equal amounts at the
end of each year. If the 15,000Frw isinvested at 4% per annum, what is the annual payout to the family?
Solution:
This example asks what annual payment is needed at 4% for 10 years to disperse
15,000Frw. That is, we can think of the 15,000Frw as a loan amortized at 4%
for 10 years. Thepayment needed to pay off the loan is the yearly amount Mr.
Rwambikana’s family willreceive.The yearly payout P is
3) Mr Unen is 20 years away from retiring and starts saving $100 a month in
an accountpaying 6% compounded monthly. When he retires, he wishes to
withdraw a fixedamount each month for 25 years. What will this fixed amountbe?
APPLICATION ACTIVITY 2.3.5
A corporation is faced with a choice between two machines, both of
which are designedto improve operations by saving on labor costs.
Machine A costs $8000 and will generatean annual labor savings of
$2000. Machine B costs $6000 and will save $1800 in laborannually.
Machine A has a useful life of 7 years while machine B has a useful
life of only5 years. Assuming that the time value of money (the
investment opportunity rate) of the corporation is 10% per annum,
which machine is preferable? (Assume annual compounding andthat the savings is realized at the end of each year).
