Exponential models: growth, decay and logarithmic scales · Grade 12 · 5 units
In Exponential models: growth, decay and logarithmic scales (Grade 12 · 5 units) you practise 5 skills: Exponential growth · The half life · Logarithmic scales · An accumulated amount · Comparing two cultures. In exponential growth the amount is multiplied by the same number in every equal period. When the number of bacteria doubles every hour, after some hours it is the starting number times two to the power of the number of hours, and one hour earlier it was half of what it is now. In half an hour the number grows by a factor of the square root of two, because two half hours together make one doubling.
In exponential decay the amount halves in each fixed period, called the half life. When the amount is a number times e to the power of minus k t, the half life is the natural log of two divided by k. This is how you work out how much of a medicine is left in the blood, and how long it takes for half of it to be left. On a logarithmic scale each step of the scale is a multiplication by the same number. In a binary search the number of steps is the base two logarithm of the number of words, rounded up, on the Richter scale each rise of one multiplies the amplitude by ten, and in decibels each rise of ten multiplies the intensity by ten. When a rate is given, the amount accumulated over a period is the integral of the rate over that period.
Tip: The half life does not depend on the starting amount: from any amount, half is left after the same time.
What you learn
Exponential growth
For example: A culture starts with 1,000 bacteria, and their number doubles every hour. The table shows the first hours.
| Hour | Bacteria |
|---|---|
| 0 | 1,000 |
| 1 | 2,000 |
| 2 | 4,000 |
Answer and solution
The answer: 32000
After 5 hours there are 1000 × 2⁵ = 32,000 bacteria.
The half life
For example: There are 4 milligrams of a medicine in the blood, and the amount halves every 4 hours. How many milligrams are left after 4 hours?
Answer and solution
The answer: 2
The number of periods is 4 ÷ 4 = 1 , so 4 ÷ 2 = 2 milligrams are left.
Logarithmic scales
For example: In a binary search of a sorted dictionary, each step compares the word you want with the middle word and rules out half of the words left, until one word remains. How many steps does a dictionary of 2 words need?
Answer and solution
The answer: 1
The number of steps is log₂ 2 = 1 , because 2¹ = 2 .
An accumulated amount
For example: The rate of gold production in the world, in tonnes per year, is about f(t) = 1500 × e^(0.05t) , where t is the number of years from the start of the period. What is the rate at the start of the period, in tonnes per year?
Answer and solution
The answer: 1500
At the start of the period substitute t = 0 : f(0) = 1500 × e⁰ = 1500 .
Comparing two cultures
For example: Two dishes hold cultures of bacteria. In dish A the number of bacteria grows by a factor of a every hour, and in dish B by a factor of b . In three hours the number in dish A grows by a factor of 64 and the number in dish B by a factor of 8 . What is the relation between a and b ?
- b = a²
- a = b
- a = b³
- a = b²
Answer and solution
The answer: a = b²
Here a³ = 64 , so a = 4 , and b³ = 8 , so b = 2 . Hence a = b² .
Not on the site yet
Parts of this topic that are not built yet: Reading parameters from a graph on a logarithmic scale · Comparing medicines in an applet.
The exercises are made by rules and the answers are worked out in code, so there are new exercises every time. The topic follows the Israeli Ministry of Education’s high school maths curriculum at 5 units (grades 10 to 12).