Albert Barlett Exponential Function

The greatest shortcoming of the human race is our inability to understand the exponential function.

The Exponential function is used to describe the size of anything that’s growing steadily, for example, 5% per year.

We are talking about a situation where the time that is required for the growing quantity to increase by a fixed fraction is constant.

If it takes a fixed length of time to grow 5%, then it follows that it takes a longer fixed length of time to grow by 100%. This longer time is called the doubling time.

We can calculate the doubling time:
T2 = 70 / (% growth per unit time)
Thus, a growing rate of 5% per year has a doubling time of:
T2 = 70 / 5 = 14 years

Doubling time for 7% growth = 10 years.

In just 10 doubling times it’s a thousand times larger than when it started.

The growth in any doubling time is greater than the total of all the preceding growth!

Steady Growth for 70 Years:

Growth Rate                           Factor
1% per year                            2 = 2
2%                                 2 x 2 = 4
3%                             2 x 2 x 2 = 8
4%                        2 x 2 x 2 x 2 = 16
…
7%                                 2^7 = 128

Imagine bacteria growing steadily in a bottle. They double in number every minute.

At 11:00 am there is one bacteria in the bottle.
At 12:00 noon the bottle is full.

11:54 am                1/64 = 1.6% full
11:55 am                1/32 = 3.1% full
11:56 am                1/16 = 6.3% full
11:57 am                1/8 = 12.5% full
11:58 am                  1/4 = 25% full
11:59 am                  1/2 = 50% full
12:00 noon                     100% full
12:01 pm                       200% full
12:02 pm                       400% full

We cannot let others do our thinking for us.

Growth of populations and growth of rates of consumption of resources can not be sustained.

Central to the things that we must do is to recognize that population growth is the immediate cause of all of our resource and environmental crises.

by Albert Bartlett