APPLICATIONS OF EXPONENTIAL | |

AND | |

LOGARITHMIC FUNCTIONS |

**DECAY WORD PROBLEMS:**

As with any word problem, the trick is convert a narrative statement or question to a mathematical statement.

**Example 1:**

**Solution:**

Let's start by calculation some points, graphing the points, and then determining what mathematical model best describes the points. The number of remaining cells at any time depends on the amount of time since the experiment began. All the points will have the form (time since experiment began, number of remaining cells). This sounds complicated, but it is not

For example, at the start of the experiment no time
has passed and *time*=0. At the start of the experiment there are 5,000,000 cells. Therefore the
first point is

One minute after the experiment starts, 45% of the five million cells die leaving 55% X 5,000,000 = 2,750,000 cells. The corresponding point is

Two minutes after the experiment starts, 45% of the 2,750,000 cells die leaving 55% X 2,750,000 = 1,512,500 cells. The corresponding point is

Three minutes after the experiment starts, 45% of 1,512,500 cells die leaving 55% X 1,512,500 = 831,875 cells. The corresponding point is

When you plot these points, you note that the curve looks exponential. Therefore, the mathematical model is probably exponential.

A mathematical model is nothing more than a fancy word for an equation. An exponential equation looks something like the following equation:

We know that
*a*=5,000,000 because we started with five million cells. You
can verify this in the equation
by letting
*t*=0 in the equation.

The equation is now modified:

We know that there are 2,750,000 cells after 1 minute. Another way of
saying this is that
In the above equation,
replace
with 2,750,000 and replace *t* with 1.

Take the natural logarithm of both sides of the equation:

The equation describing the number of cells remaining after an experiment has begun is

Let's check it out by seeing if this model will give us 1,512,500 cells after two minutes.

How long will it take the sample to decay to below 1,000 cells? Just substitute 1,000 for in the equation and solve for t.

Take the natural logarithm of both sides of the equation.

It will take about 14.25 minutes for the cell population to drop below a
1,000 count.

Let's check our answer:

In the mathematical model
everywhere there is a t, substitute 14.25.
If the answer is less than 1,000, our answer is correct.

**Relationship between base and decay constant:**

Suppose you have an equation
Recall that
you can rewrite this equation as

Suppose we wanted a base of 7 instead of a base of e. In order words, we
wanted our equation to look like
This
can re rewritten as
If
we want both equations to represent the same thing, we must find a value for
b such that
7^{b}=54.5981500331 or something very close to this
quantity.

Note that can be rewritten as

If you would like to test your knowledge by working some problems, click on
problem.

If you would like to go back to the table of contents, click on
contents.

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