Please Explain Variation of Poe^rt formula

[BigNate]

Hello,

I recently learned about the formula for exponential growth: P(t)=Po(e^(rt)). However, I found a review problem online that used a slight variation to that formula. They used P(t)=Po(2^(rt)). Can someone please explain this to me?

The question is: "At current growth rates, the Earth's population is doubling about every 69 years. If this growth rate were to continue, about how many years will it take for the Earth's population to increase 50% from the present level?"

The used the formula: P(t)=Po(2^(t/69))...then solved

Here are my questions:
1. Why did they not use e and instead used 2?
2. How did they get r*t to be t/69?

Thanks in advance for your help!

 

[Dr.Peterson]

I recently learned about the formula for exponential growth: P(t)=Po(e^(rt)). However, I found a review problem online that used a slight variation to that formula. They used P(t)=Po(2^(rt)). Can someone please explain this to me?

The question is: "At current growth rates, the Earth's population is doubling about every 69 years. If this growth rate were to continue, about how many years will it take for the Earth's population to increase 50% from the present level?"

The used the formula: P(t)=Po(2^(t/69))...then solved

Here are my questions:
1. Why did they not use e and instead used 2?
2. How did they get r*t to be t/69?

Thanks in advance for your help!

When e is the base, the quantity is multiplied by e when the exponent is 1; when the base is 2, it is multiplied by 2. Thus, in the formula P(t)=Po(2^(t/69)), the quantity doubles every 69 years, which is just what the problem states.

In general, P(t)=Po(2^(t/T)) represents growth with a doubling time of T.

If you have a textbook, you might look up "doubling time" in the index.

 

[tkhunny]

There are some believers in the "e" structure. Some discussions (classroom, exam expectations, etc.) simply refuse to allow any base but e. I find this attitude a bit pedantic. Obviously, not everyone agrees with my opinion. Of the simplest examples, the doubling time has already been mentioned. A base of 2 might be reasonable. In addition, the half-life problem may lend itself to the base 1/2, for reasons just as obvious. Of course, if your exam is machine-scored, and your teacher requires base e, you will need to comply for the examination. You can just smile and move on with your life.