1 MCU504/Unit 5: Exponential Number Vs Logarithmic numbers, Financial Mathematics -Mr. Randimbiarison DMHS Unit 5 (Part 1): Exponential (power) number vs logarithmic number A function often is expressed in the form of y = f(x). This notation materializes the relation between the dependent variable y and the independent variable x, i.e., y is said to be dependent of x, or y is a function of x: when x changes in value, y changes accordingly. An exponential number or generally expressed as an exponential function is represented by an equation in its basic form: ( ) = or = or = ××××××……..× 6 y is the power value or the exponential value a is the initial amount (quantity) or initial value c is the base of the exponential function (number) x is the exponent or the number of times the base c is multiplied by itself Remember that for any value c raised to the power of zero, c is always equal to 1: = 1 (Replace c by 123,456,789 on your calculator then raise it to the power of 0. What did you find?)
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1 MCU504/Unit 5: Exponential Number Vs Logarithmic numbers, Financial Mathematics -Mr. Randimbiarison DMHS
Unit 5 (Part 1): Exponential (power) number vs logarithmic number
A function often is expressed in the form of y = f(x). This notation materializes the relation between the
dependent variable y and the independent variable x, i.e., y is said to be dependent of x, or y is a function of x:
when x changes in value, y changes accordingly.
An exponential number or generally expressed as an exponential function is represented by an equation in its
𝒚 = (𝟏𝟎)(1.77815125−log 12) . 𝒚 = (𝟏𝟎)(1.77815125−log 12) . On your calculator, enter 10 ^ (1.77815125 – (log12 ÷ log 10))
or simply enter 10 ^ (1.77815125 – log12) → answer→ y = 4.999999996 or y = 5
Check: log 12x5 = log 60? = 1.77815125 (Check)
2nd Method: → 𝑙𝑜𝑔 12𝑦 = 1.77815125 →because it’s in base 10, we can write →𝟏𝟐𝑦 = (𝟏𝟎)(1.77815125) → 𝑦 = (𝟏𝟎)
(1.77815125)
𝟏𝟐 = 5
(Better approach)
On your calculator, enter (10 ^ 1.77815125) ÷ 12
11 MCU504/Unit 5: Exponential Number Vs Logarithmic numbers, Financial Mathematics -Mr. Randimbiarison DMHS
Exponential Growth (Growth is synonym of rise, increase, appreciation, augmentation, etc.)
Exponential growth models apply to any situation where the growth is proportional to the current size of the quantity of interest.
Exponential growth models are often used for real-world situations like interest earned on an investment, human or animal population, bacterial culture growth, etc.
The general exponential growth model is
→ y increases in value as t increases
where a is the initial amount or number, r is the growth rate (for example, r =2% growth rate means r=0.02 in decimal form), and t (the independent variable, same as x) is the time elapsed.
𝒚 = 𝒂(𝟏 + 𝒓)𝒕
12 MCU504/Unit 5: Exponential Number Vs Logarithmic numbers, Financial Mathematics -Mr. Randimbiarison DMHS
Example 1: Angelstown has a population of 32,000. It grows with a 5% annual rate. This situation can be modeled by the
equation:
𝒚 = 𝟑𝟐𝟎𝟎𝟎(𝟏 + 𝟎. 𝟎𝟓)𝒕 (Convert the rate to decimal value: Rate 5% = 5
100 = 5÷100 = 0.05)
or 𝒚 = 𝟑𝟐𝟎𝟎𝟎(𝟏. 𝟎𝟓)𝒕, with t the number of years.
Question: After how many years (t) will the population of Angeltown reaches 140,000?
On your calculator, enter the following: log (140000÷32000) ÷ log1.05
13 MCU504/Unit 5: Exponential Number Vs Logarithmic numbers, Financial Mathematics -Mr. Randimbiarison DMHS
Sometimes, you may be given a doubling or tripling rate rather than a growth rate in percent (%).
For example, if you are told that the number of cells in a bacterial culture doubles every hour, then the equation to model the
situation would be:
Doubling rate: 𝒚 = 𝒂(𝟐)𝒕 or tripling rate: 𝒚 = 𝒂(𝟑)𝒕 with t is the number of hours elapsed, a is the initial quantity.
Example 2: Suppose a culture of 100 bacteria is put into a petri dish and the culture doubles in size every hour. Predict the number of bacteria that will be in the dish after 12 hours. Solution:
𝒚 = 𝒇(𝒕) = 𝒂(𝟐)𝒕 where a = 100 bacteria, the base a = 2, t is the time in hours
𝒚 = 𝟏𝟎𝟎(𝟐)𝟏𝟐
Enter the following on your calculator: 100 x (2^12) → answer → y = 409,600 bacteria
14 MCU504/Unit 5: Exponential Number Vs Logarithmic numbers, Financial Mathematics -Mr. Randimbiarison DMHS
Example 3:
Suppose a culture of 20 bacteria is put into a petri dish and the culture triples in size every hour. After how many hours does it attain a number of 1,200,000 bacteria. Solution:
𝒚 = 𝒇(𝒕) = 𝒂(𝟑)𝒕 where a = 20 and y = 1,200,00 bacteria after t hours. t?
𝟏, 𝟐𝟎𝟎, 𝟎𝟎𝟎 = 𝟐𝟎(𝟑)𝒕
→ 1,200,000 = 20(3)𝑡
→ by dividing by 20 on both sides 1,200,000
20= (3)𝑡
→ Then calculate 𝑡 =log (
1,200,000
20)
𝑙𝑜𝑔 3
→ Enter the following on your calculator: 𝐥𝐨𝐠 (𝟏 𝟐𝟎𝟎 𝟎𝟎𝟎 ÷ 𝟐𝟎) ÷ 𝒍𝒐𝒈 𝟑 → answer: t = 10 hours
15 MCU504/Unit 5: Exponential Number Vs Logarithmic numbers, Financial Mathematics -Mr. Randimbiarison DMHS
Exponential Decay (Decay is synonym of decline, decrease, depreciation, return, etc.)
Exponential decay models apply to any situation where the decay is proportional to the current size of the quantity of interest. In other words, the value of the initial quantity decreases over time.
Exponential decay models are often used for real-world situations such as depreciation (or decrease) of the price of a car, or a decrease of animal population of endangered species, etc.
The general exponential decay model is
→ y decreases in value as t increases
where a is the initial amount(quantity) or initial value, r is the rate of decay (for example, r =0.5% decay rate r=0.005 in decimal form),
and t (or x) is the time elapsed.
Note that the Rate of 0.5% = 0.5
100 = 0.5÷100 = 0.005
Example 1:
You bought a second hand 2018 Honda Civic for $19,690 from a Honda dealer. Your car will depreciate in value by a rate of 11% annually. What will be the resale price value of your car in 10 years? Solution:
𝒚 = 𝒂(𝟏 − 𝒓)𝒕 where a = 19690, r = 11% or r = 0.11, and t = 10 years
𝒚 = 𝟏𝟗𝟔𝟗𝟎(𝟏 − 𝟎. 𝟏𝟏)𝟏𝟎 → y = 6139.68
On your calculator, enter: 19690 x (1 – 0.11) ^10 → answer = 6139.680654
Conclusion: Your car is worth $ 6139.68 in 10 years
𝒚 = 𝒂(𝟏 − 𝒓)𝒕
16 MCU504/Unit 5: Exponential Number Vs Logarithmic numbers, Financial Mathematics -Mr. Randimbiarison DMHS
Example 2:
Sandra bought a second hand 2017 Subaru Forester for $22,500. She wants to know when will her car worth about $10,000 if the depreciation price rate is 7% every year.
Solution:
𝒚 = 𝒂(𝟏 − 𝒓)𝒕 where a = 22,500, r = 7% or r = 0.07, y = 10,000: t =?
𝟏𝟎𝟎𝟎𝟎 = 𝟐𝟐𝟓𝟎𝟎(𝟏 − 𝟎. 𝟎𝟕)𝒕 → 𝑡 =log (
10000
22500)
𝑙𝑜𝑔 (1−0.07)
→ Enter the following on your calculator: log (10000÷22500) ÷ 𝐥𝐨𝐠 (𝟏 − 𝟎. 𝟎𝟕)
→ 𝑡 = 11.17
Conclusion: Approximately in 11 years your car is worth $10,000
17 MCU504/Unit 5: Exponential Number Vs Logarithmic numbers, Financial Mathematics -Mr. Randimbiarison DMHS
Example 3:
Newfoundland cod stocks suffer serious and surprising decline. About 12,000 tonnes of cod were harvested in 2018. The cod population continues to decline at an alarming average rate of 6% annually. Approximately, how many tonnes of cod fish were harvested in 2008 knowing that this issue has been recorded in the past three decades?
Solution:
𝒚 = 𝒂(𝟏 − 𝒓)𝒕 where y = 12,000, r = 6% or r = 0.06, t =2018 – 2008 = 10 years: a =?
𝟏𝟐𝟎𝟎𝟎 = 𝒂(𝟏 − 𝟎. 𝟎𝟔)𝟏𝟎 , isolate 𝒂 by dividing by (𝟏 − 𝟎. 𝟎𝟔)𝟏𝟎on both sides
12000
(𝟏−𝟎.𝟎𝟔)𝟏𝟎=
𝒂(𝟏−𝟎.𝟎𝟔)𝟏𝟎
(𝟏−𝟎.𝟎𝟔)𝟏𝟎 →
12000
(𝟏−𝟎.𝟎𝟔)𝟏𝟎= 𝑎
→ Enter the following on your calculator: 12000÷ ((1 - 0.06) ^10) → 𝑎 = 22,279.35
Conclusion: Approximately 22,280 tonnes of cod fish were harvested in 2008
18 MCU504/Unit 5: Exponential Number Vs Logarithmic numbers, Financial Mathematics -Mr. Randimbiarison DMHS
Very Important note: When will the rule in the form of 𝒚 = 𝒂(𝒄)𝒃𝒕 𝒐𝒓 𝒚 = 𝒂(𝒄)𝒃𝒙 be used?
Example 1: Suppose you would like to study the growth of a culture of 100 bacteria. We saw how you would write the rule
when the culture doubles in size every hour, 𝒚 = 𝒂(𝒄)𝒕, where b=1, since the growth is on every unit of time.
Question: Predict the number of bacteria that will be in the petri dish after 12 hours in the following scenarios:
The bacteria: a) doubles in size every half hour? b) doubles in size every two hours? c) double in size every two days?
Some people may “intuitively” write the following rules for each scenario:
a) 𝒚 = 𝟏𝟎𝟎(𝟐)𝟎.𝟓𝒕 , t = # hours
b) 𝒚 = 𝟏𝟎𝟎(𝟐)𝟐𝒕 , t = #hours
c) 𝒚 = 𝟏𝟎𝟎(𝟐)𝟐𝒕 , t = #days The rules a), b), c) above are wrong.
19 MCU504/Unit 5: Exponential Number Vs Logarithmic numbers, Financial Mathematics -Mr. Randimbiarison DMHS
Solution: Although it is possible to write an exponential rule by intuition, the safest way to find it is by using the following steps:
1) Corner the unit of time t in question. Use the time unit that describes how often the base c is multiplied by itself. 2) Use a small table of value and pay attention to the pattern for each value of t.
3) Calculate the value of the coefficient b in the general equation 𝒚 = 𝒂(𝒄)𝒃𝒕
a) The bacteria doubles in size every half hour? Initial amount a = 100 bacteria
t = # hours 𝒚 = 𝒂(𝒄)𝒃𝒕 y You can continue, if you want, and enter more
values of t in the table. The two entries are enough to determine the value of b.
0 𝒚 = 𝟏𝟎𝟎(𝟐)𝒃.𝟎 100
0.5 𝒚 = 𝟏𝟎𝟎(𝟐)𝒃.(𝟎.𝟓) 200 (doubled every half hour)
When t = 0.5 y=200 → 200 = 100(2)𝑏(0.5)or 200 = 100(2)0.5𝑏
Because, I want to isolate the term with the unknown value b, divide both sides by 100 → 𝟐𝟎𝟎
𝟏𝟎𝟎=
𝟏𝟎𝟎(𝟐)𝟎.𝟓𝒃
𝟏𝟎𝟎
→ 2 = (2)0.5𝑏
Using logarithm number → 0.5b = 𝑙𝑜𝑔 2
𝑙𝑜𝑔 2 = 1 → 0.5b = 1 → b = 2
Therefore, the rule of scenario a) is 𝒚 = 𝟏𝟎𝟎(𝟐)𝟐𝒕
Conclusion: In 12 hours the number of bacteria will be 𝒚 = 𝟏𝟎𝟎(𝟐)𝟐(𝟏𝟐)= 𝒚 = 𝟏𝟎𝟎(𝟐)𝟐𝟒 → y = 1,677,721,600 (approximately, over a billion and a half bacteria)
20 MCU504/Unit 5: Exponential Number Vs Logarithmic numbers, Financial Mathematics -Mr. Randimbiarison DMHS
b) The bacteria doubles in size every two hours? Initial amount a = 100 bacteria
t = # hours 𝒚 = 𝒂(𝒄)𝒃.𝒕 y
0 𝒚 = 𝟏𝟎𝟎(𝟐)𝒃.𝟎 100
2 𝒚 = 𝟏𝟎𝟎(𝟐)𝒃.𝟐 200 (doubled every two hours)
When t = 2 y=200 → 200 = 100(2)𝑏(2)or 200 = 100(2)2𝑏
To determine b, isolate the term with the unknown value b, divide by 100 on both sides → 𝟐𝟎𝟎
𝟏𝟎𝟎=
𝟏𝟎𝟎(𝟐)𝟐𝒃
𝟏𝟎𝟎
→ 2 = (2)2𝑏
Using logarithm number → 2b = 𝑙𝑜𝑔 2
𝑙𝑜𝑔 2 = 1 → 2b = 1 → b =
1
2
Therefore, the rule of scenario b) is 𝒚 = 𝟏𝟎𝟎(𝟐)𝟏
𝟐𝒕 or 𝒚 = 𝟏𝟎𝟎(𝟐)
𝒕
𝟐
Conclusion: In 12 hours the number of bacteria will be 𝒚 = 𝟏𝟎𝟎(𝟐)𝟏𝟐
𝟐 = 𝒚 = 𝟏𝟎𝟎(𝟐)𝟔 → y = 6400
21 MCU504/Unit 5: Exponential Number Vs Logarithmic numbers, Financial Mathematics -Mr. Randimbiarison DMHS
c) The bacteria doubles in size every two days? Initial amount a = 100 bacteria
t = # hours 𝒚 = 𝒂(𝒄)𝒃.𝒕 y 2 days = 48 hours
0 𝒚 = 𝟏𝟎𝟎(𝟐)𝒃.𝟎 100
48 𝒚 = 𝟏𝟎𝟎(𝟐)𝒃.𝟒𝟖 200 (doubled every two hours)
When t = 48, y=200 → 200 = 100(2)𝑏(48)or 200 = 100(2)48𝑏
To determine b, isolate the term with the unknown value b, divide by 100 on both sides → 𝟐𝟎𝟎
𝟏𝟎𝟎=
𝟏𝟎𝟎(𝟐)𝟒𝟖𝒃
𝟏𝟎𝟎
→ 2 = (2)48𝑏
Using logarithm number → 48b = 𝑙𝑜𝑔 2
𝑙𝑜𝑔 2 = 1 → 48b = 1 → b =
1
48
Therefore, the rule of scenario c) is 𝒚 = 𝟏𝟎𝟎(𝟐)𝟏
𝟒𝟖𝒕 or 𝒚 = 𝟏𝟎𝟎(𝟐)
𝒕
𝟒𝟖 On your calculator, enter: 100 x (2^ (1÷4))
Conclusion: In 12 hours the number of bacteria will be 𝒚 = 𝟏𝟎𝟎(𝟐)𝟏𝟐
𝟒𝟖 = 𝒚 = 𝟏𝟎𝟎(𝟐)𝟏
𝟒 → approximately y = 118 bacteria
*Another way of solving scenario c) would be to choose the time unit as t = # days:
t = # days 𝒚 = 𝒂(𝒄)𝒃.𝒕 y When t = 2, y=200 → 200 = 100(2)𝑏.2
or 200 = 100(2)2𝑏 → 2b = 𝑙𝑜𝑔 2
𝑙𝑜𝑔 2 → b =
𝟏
𝟐
The rule is 𝒚 = 𝟏𝟎𝟎(𝟐)𝟏
𝟐𝒕
t=12 hours = 0.5 day → 𝒚 = 𝟏𝟎𝟎(𝟐)𝟏
𝟐(𝟎.𝟓)or 𝒚 = 𝟏𝟎𝟎(𝟐)
𝟏
𝟒
0 𝒚 = 𝟏𝟎𝟎(𝟐)𝒃.𝟎 100
2 𝒚 = 𝟏𝟎𝟎(𝟐)𝒃.𝟐 200 (doubled every two days)
The answer will be the same: In 12 hours or 0.5 days the bacteria grows up to about 118.