Table of Contents

• Section 1. Exact Doubling Time
• 1: In search of ... the exact doubling time
• 2: Finding doubling time with general numbers
• 3: Logarithms = Exponent-busters
• 4: Test your knowledge of logs
• 5: More exponent busting
• Section 2. Applications
• 6: World Population
• 7: Growth of technology
• 8: How long before the oysters are unsafe?
• Review
• 9: Summary

Finding doubling time with general numbers

Recall that we started out with an equation:

N_t = N₀ 2ⁿ

Remember the exponential part of this equation is that we raise 2 to the power of the number of generations, n. Another way of saying this is that, for every generation, we multiply the original population by 2.

Let's say I told you that N₀ was 10 million and N_t was 80 million one hour later. Luckily for you, these are nice neat numbers. You would probably think to yourself, "well, the population doubled three times, from 10 to 20 million, from 20 to 40 million, and from 40 to 80 million. Since three doublings took 1 hour, each doubling took 20 minutes."

Another somewhat fancier way of getting the same answer would be to substitute the information that you know (N_t and N₀) into the equation above, and solve for the information that you don't know (number of generations, n). Here's how that would look with some conveniently chosen special numbers:

 

 

Let's try the same basic procedure with more general numbers. Let's say the population increased from 10 to 70 million in one hour.

 

We have not solved our problem yet, and what's in the way is that 2 is raised to some power in order to get 7 -- but we don't have a way to calculate what that power is. In other words, the exponent is the trouble. We need a way to get the "exponential" out of "exponential growth" so we can deal with the maths. But how?