Exponential & Logarithmic Models — Practice
Growth and decay that depend on how much is already there, and the logs that solve them.
What these questions cover
These questions use 5 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Ratio — A comparison of two quantities, like 3 cups flour to 2 cups water.
- Factor — A number that divides evenly into another.
- Growth factor — b itself. b > 1 means growth, 0 < b < 1 means decay.
- Half-life — The time for a decaying quantity to fall to half.
- Magnitude — The length of the vector, written |v|.
Problem 1
Is the table 4, 8, 16, 32 linear or exponential?
- Exponential
- Linear
- Neither
- Both
Worked answer
Exponential
Each is double the last
A constant ratio means exponential
Problem 2
Is the table 4, 9, 14, 19 linear or exponential?
- Linear
- Exponential
- Neither
- Both
Worked answer
Linear
Each adds 5
A constant difference means linear
Problem 3
A quantity grows 6% a year. What is the growth factor?
Worked answer
1.06
1+0.06=1.06
Problem 4
A quantity falls 25% a year. What is the decay factor?
Worked answer
0.75
1-0.25=0.75
Problem 5
For y=100(2)t, find y when t=3.
Worked answer
800
23=8
100 × 8=800
Problem 6
A town of 5000 grows 4% a year. Find the population after 2 years, to the nearest whole number.
Worked answer
5408
1.042=1.0816
5000 × 1.0816=5408
Problem 7
A 64 g sample halves every 3 days. How much remains after 12 days?
Worked answer
4
12 ÷ 3 = 4 halvings
64 → 32 → 16 → 8 → 4
Problem 8
A 200 mg dose halves every 6 hours. How much remains after 18 hours?
Worked answer
25
18 ÷ 6 = 3
200 → 100 → 50 → 25
Problem 9
Write the growth factor for a quantity that triples each period.
Worked answer
3
b=3
Problem 10
$500 at 8% compounded annually. Find the value after 3 years, to the nearest dollar.
Worked answer
630
1.083 ≈ 1.2597
500 × 1.2597 ≈ 630
Problem 11
A population falls 30% a year. Which model is right?
- y=a(0.7)t
- y=a(0.3)t
- y=a(1.3)t
- y=a(-0.3)t
Worked answer
y=a(0.7)t
70% survives
So the factor is 0.7
Problem 12
Solve 2t = 40 for t, to two decimal places.
Worked answer
5.32
t=log 40log 2
≈ 5.32
Problem 13
How many years for $1000 at 5% to double? Give the answer to one decimal place.
Worked answer
14.2
t=log 2log 1.05
≈ 14.2
Problem 14
A sample of 80 g has half-life 10 years. How many grams remain after 40 years?
Worked answer
5
80 → 40 → 20 → 10 → 5
Problem 15
A magnitude 8 earthquake releases how many times the energy of a magnitude 6?
- About 1000
- About 2
- About 100
- About 32
Worked answer
About 1000
Two steps
32 × 32 ≈ 1000
Problem 16
A culture doubles every 4 hours and now has 1600 cells. How many were there 12 hours ago?
Worked answer
200
12 ÷ 4 = 3 doublings
1600 ÷ 8 = 200