- A tree frog population doubles every three weeks. Suppose that currently, there are 10 tree frogs in your back yard. How many tree frogs will there be in 6 months, assuming that there are four weeks each month?
- How long will it take this population to be 10,240?
- We’ll first have to figure out how many times this population will double in 4 months. With four weeks each month, the six months will be equal to 24 weeks. Since the population will double every three weeks, it will double 8 times in 24 weeks. Let’s look at the table below:
Weeks doubling periods Population
0 0 10
3 1 20 = 10 x 2, or 10 x 21
6 2 40 = 10 x 2 x 2, or 10 x 22
9 3 80 = 10 x 2 x 2 x 2, or 10 x 23
After 24 weeks, the population will be 10 x 28, or 2560 tree frogs!
- If you look at the table above, you will notice that you could let n be the number of weeks. How would you get the number of doubling periods from n?
So we need to solve the equation 10,240 = 10 x 2(n/3). If you divide both sides by 10, you’ll have 1024 = 2(n/3).
With some thought, you can recall that 1024 = 210. Watch what results:
1024 = 2(n/3)
210 = 2(n/3).
So how can you find n?
Think hard! Both sides of the equation are written as powers of 2, so what has to be true? Yes, the exponents, have to be the same!
So now we know that 10 = n/3, and n = 30. That means that after 30 weeks, the population will be 10,240.