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An Introduction to Exponential Growth

Date: 10/07/2006 at 14:23:22
From: Sara
Subject: exponential growth?


What exactly is exponential growth?  At this moment I am using my 
book's study aid and all I see is equations and a lot of exponents. 
Please tell me what they are and what they mean.

Thank you!

Date: 10/09/2006 at 15:40:27
From: Doctor Ian
Subject: Re: exponential growth?

Hi Sara,

It's helpful to think about other kinds of growth.  For example,
something might grow linearly, e.g., 

  y = 2x

In this case, for a given change in x, we get the same change in y, no
matter what x is.  In the case above, changing x from 5 to 6 gives us
a difference of 

   2*6 - 2*5 = 2*(6 - 5) = 2*1 = 2

A change of 1005 to 1006 gives the same difference:

   2*1006 - 2*1005 = 2*(1006 - 1005) = 2*1 = 2

If we plot all the (x,y) pairs that satisfy this kind of growth, they
all lie on a line, which is why we say the growth is "linear".  

Does that make sense so far?  Next, consider something like

  y = x^2

Here, we would say that the growth is "polynomial", because we have a
variable and constant exponents greater than 1.  In this case, the
change in y corresponding to a given change in x depends on the value
of x.  Go ahead and try evaluating that the way I did above. 

In the case above, plotting these points would give you a parabola,
which increases more steeply the farther x gets from the origin.  

Now, suppose that instead of having a variable with a constant
exponent, we have a constant with a variable exponent?  That might
look like this:

  y = 2^x

When the exponent is the thing that's changing, we say that the growth
is "exponential".

Near the origin, this looks a lot like the parabola, but it grows
_much_ more quickly than any polynomial.  To see how much more
quickly, try plotting all three functions,

  y = 2x

  y = x^2

  y = 2^x

using a graphing calculator.  Or, you can make a table, with different
values for x:

    x      2x    x^2      2^x
  -----  -----  -----  ---------
      1      2      1          2
      2      4      4          4
      3      6      9          8
      4      8     16         16
      5     10     25         32
     10     20    100      1,024
     20     40    400  1,048,576
When the values of x are small, there isn't a lot of difference; but
as the value of x gets large, exponential growth overwhelms linear and
polynomial growth. 

This has two consequences that are worth knowing about.  

One is that when we've identified true exponential growth, it's
usually something we should be interested in--whether it's a good
thing (e.g., a growth in profits) or a bad thing (e.g., a growth in
the number of cases of a disease).  

The other is that because "exponential growth" sounds so dramatic,
many people say that something is growing exponentially when they just
mean it's growing quickly.  It's a kind of exaggeration, so when you
hear someone talking about exponential growth, you need to seriously
consider which of the following is true:  (1) the person is right, (2)
the person is wrong, but doesn't realize it, or (3) the person is
wrong and realizes it, but hopes that you don't. 

Does this make sense?  Let me know if you need more help. 

- Doctor Ian, The Math Forum 
Associated Topics:
High School Exponents
High School Statistics

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