Alternative Formulas for Growth and Decay

Date: 08/02/2002 at 21:59:38
From: Jared
Subject: Math Analysis/Algebra 2

I was just wondering about the Growth and Decay formula. Last year my
teacher told me it was 

  q = q_0 * e^(rt)

My teacher this year says it's 

  q = q_0 * a^t 

One of these says that the variable r decides the growth and decay and
the other say that the variable a decides the growth or decay. And why
does one equation include e and the other doesn't? I'm confused.

Date: 08/02/2002 at 23:41:51
From: Doctor Peterson
Subject: Re: Math Analysis/Algebra 2

Hi, Jared.

These two equations are equivalent; it's sort of like saying that you 
can measure your height in either feet or meters.

Since we can write a as e^ln(a), the second equation is the same as

    q = q_0 e^(ln(a)t)

and the r from the first equation is just ln(a).

Which form you use  depends on your point of view. The first form 
emphasizes the time constant r, which tells how long it takes to grow 
to e times the initial amount. This is the more sophisticated form, I 
suppose; the exponential with e makes calculus easier, such as 
finding the instantaneous rate of increase. 

The second form emphasizes the base a, which is how much you grow by
in a unit of time. This links it to geometrical sequences. 

Another useful form is 2^(-t/h), where h is the half-life, the time it
takes to reach 1/2 the original amount. Each of these has its own
special use.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 08/04/2002 at 19:40:44
From: Jared
Subject: Thank you (Math Analysis/Algebra 2)

Thanks, I understand it now... you really helped!